Intersection in low dimensional topologyIntersection in low dimensional topology Vincent Florens Habilitation a diriger des recherches Laboratoire LMAP UPPA { Universit e de Pau et

Intersection in low dimensional topology

Vincent Florens

Habilitation a diriger des recherches

Laboratoire LMAPUPPA – Universite de Pau et des Pays de l’Adour

V.Kandinsky. Circles in a Circle

JURYARTAL Enrique, Universite de Saragosse

BOILEAU Michel, Universite de Aix-MarseilleDEGTYAREV Alex, Universite de Bilkent

FIEDLER Thomas, Universite Paul SabatierLESCOP Christine, Universite de Grenoble Alpes (Presidente)

OREVKOV Stepan, Universite Paul SabatierVALLES Jean, Universite de Pau et des pays de l’Adour

RAPPORTEURSBLANCHET Chistian (Paris), LIBGOBER Anatoly (Chicago), SUCIU Alex (Boston)

Contents

Introduction 31. Bibliography of the author 72. List of co-authors 83. Structure of the memoire 9

Chapter 1. Preliminaries 111. Twisted homology 112. Intersection forms 123. Characteristic varieties and Alexander invariants 134. Combinatorial torsion 14

Chapter 2. Surfaces in 4-manifolds and colored links 171. Signature of colored links 182. Concordance and slice genus 273. Signature of a splice 294. Perspectives 34

Chapter 3. Plane algebraic curves and line arrangements 411. Braid monodromy and the homotopy type 422. Alexander polynomials 433. Line arrangements 474. Perspectives 57

Chapter 4. Functorial extensions of the abelian Reidemeister torsion 611. Manifolds and categories 632. The Reidemeister functor 663. Alexander polynomials 714. The Magnus functor 765. A first step in dimension 4 816. Perspectives 84References 85

Bibliography 85

1

2 CONTENTS

Remerciements

Je voudrais tout d’abord remercier Christian Blanchet, Anatoly Libgober et Alex Suciu d’avoiraccepte de rapporter ce memoire. Je remercie tout aussi chaleureusement Enrique Artal, MichelBoileau, Alex Degtyarev, Thomas Fiedler, Christine Lescop, Stepan Orevkov, et Jean Valles d’avoirparticipe au jury.Merci a mes collegues et amis Charles, Daniele, Isabelle, Jean, et les autres pour tous ces momentset discussions, et pour le plaisir de travailler ensemble.Merci a Emmanuel, Jean-Baptiste et Paolo pour cette belle aventure qu’est Winterbraids. Longuevie a notre ecole.Une pensee pour ma famille et mes parents, pour Simin avec qui l’aventure ne fait que commencer,et pour Sacha et son appetit de vivre ainsi que la petite qui manifeste deja beaucoup d’energiedepuis le ventre de sa maman.

INTRODUCTION 3

En geometrie, il n’y a pas de chemin reserve aux rois.Euclide

Introduction

My research activities are devoted to interactions between geometric topology and algebraicgeometry. The works of Klein or Poincare illustrate their long common history and advancement.My aim is to use and develop combinatorial models from refined intersection theory in order tostudy problems in both topology of 3 or 4-manifolds and knot theory, and plane algebraic curveswith their singularities. The approach is inspired by the results of Arnold and Rokhlin and involvestechniques from Reidemeister torsion and intersection in homology, braid and mapping class groups,and configuration spaces. These techniques offer new perspectives and generalizations of classicalinvariants of links or algebraic curves related to the fundamental group of their complement, suchas the characteristic varieties and the Alexander invariants.

Intersections and linkings in manifolds (consequences of Poincare duality) are fundamental ob-jects in topology. They were a substantial motivation for Seifert to introduce the manifolds thatnow bear his name, they are at the origin of many invariants of manifolds and kept many out-standing mathematicians, among which Reidemeister, Whitney, Wall, Milnor, Freedman, strugglingwith their properties, both at the topological and algebraic levels. At the most elementary level,intersection pairings encode algebraic intersections of cycles. Any oriented 4-manifold carries anintersection pairing on its middle homology groups, that is related to a linking form on the boun-dary. Algebraically, this is a bilinear form on a free abelian group, of finite rank if the manifold iscompact. Intersection pairings generalize also to manifolds with properly discontinuous action. Thecombination of existing theories on intersections and linkings with homotopy theory was successfullyapplied in the 1960’s to highly connected manifolds of dimension > 4. For instance, the intersectionpairing determines the diffeomorphism class of any n-connected closed differentiable 2n-manifold.In dimensions less than or equal to 4, it is known that an analogous result cannot be expected. Theclosest one can get is the following result (due to Whitehead, Milnor and Wall): the isomorphismclass of the intersection form of an oriented compact simply connected 4-manifold coincides withboth its positive homotopy class and its h-cobordism class. Wall introduced a stabilization trick toshow that two manifolds with the properties above are stably diffeomorphic (i.e. become diffeomor-phic after adding by connected sum a number of copies of products S2 × S2) if and only if theirintersection form are stably isomorphic (i.e. become isomorphic after adding by orthogonal sum anumber of standard hyperbolic forms). After Donaldson’s work in the 1980’s, we know that thisresult is the best one can expect. Indeed the stabilization is effectively necessary.

4 CONTENTS

A fundamental class of 4-manifolds we consider arises as complements of plane complex algebraiccurves. Historically, the first significant results that have been obtained with the help of intersectionpairings are related to the Hilbert’s celebrated 16th problem which requires a classification of the ovalarrangements of real non-singular algebraic curves in RP 2, up to isotopy. The complete answer isonly known for the degrees ≤ 7. Arnold [Arn71] obtained necessary conditions from the arithmeticof intersection forms in the double covering of CP 2 branched along the complexification of thecurve. Since then, the question of the existence of such an algebraic curve with prescribed topologywas related to the study of topological properties of surfaces in given 4-manifolds (see Rokhlin,Viro, Kharlamov). Later, Orevkov [Ore99] made this approach more algorithmic and systematicby using braid monodromy on real curves. In particular, he observed that the quasipositivity ofa certain braid provides a new necessary condition (and this condition is equivalent in the case ofJ-holomorphic curves). It implies that the link in S3 obtained as the closure of the braid boundsa surface with given Betti numbers, smoothly and properly embedded in the ball B4. This givesan important motivation to develop necessary conditions (or extend previous existing) in terms oflink invariants. This history is in fact parallel to knot theory and questions on concordance. Inthe 20’s, Artin gave examples of smooth 2-spheres in S4, whose cross section are non-trivial knots.A natural problem was then to identify which knots can occur as such slices of spheres, or linksas slices of given surfaces. About fifty years later, using Seifert forms or Reidemeister torsion andintersection pairings, Murasugi and Fox and Milnor obtained some necessary conditions and thefirst counter-examples.

From a larger point of view and slightly different questions, the study of singular plane complexalgebraic surfaces, their moduli spaces and the stratification of their discriminants was initiatedby Enriques and Zariski in the 20’s. Their main idea was to generalize Riemann’s classical workfor Riemann surfaces of a multivalued function. In order to understand the topology of a complexprojective manifold, the best way is to project this surface onto the complex projective space (ofthe same dimension) and, then, interpret the manifold as a ramified covering along the discriminantlocus of the projection (a hypersurface). A main step in this process is to compute the fundamentalgroup of the complement of the hypersurface. In fact, as a consequence of Zariski-Lefschetz theory,by taking generic sections it is enough to study this fundamental group in the case of surfaces, i.e.,for complements of curves. Zariski and van Kampen developed their well-known method which hasbeen extensively used since then.

As a natural invariant, the fundamental group of a manifold or a knot complement contains muchtopological information. For knots, it was introduced by Dyck [Dyc82], following Cayley [Car03],and mainly developped by Wirtinger [Wir27] in the case of knots (the aim of Wirtinger was tocompute the group for algebraic links). From the successive results of Dehn [Deh10, Deh14], Fox[Fox62b] and later Papakyriakopoulos [Pap57a, Pap57b] on the peripheral system, Waldhausen[Wad94] showed that it is strong enough to detemine the knot. One interesting feature of Wirtinger’spresentation: the 2-complex associated to the presentation has the homotopy type of the complementof the link. The case of curve complements provides also many challenging open problems, as thecaracterization of their fundamental group or the description of its algebraic structure. Since thework of Zariski, it remains a mysterious and significant invariant of curves, as it was mentionedby Artin and Mazur in their survey of Zariski’s work. See the survey of Libgober [Lib07], and

INTRODUCTION 5

the survey of Artal-Cogolludo-Tokunaga [ABCT08]. However, since there is no classification offundamental groups of link or curve complement, and the isomorphism problem is undecidable,one cannot directly use the group in an effective way. Metabelian invariants such as Alexanderpolynomials or characteristic varieties are more manageable and also sensitive to many algebraicor geometric properties, as the 3 and 4-genera of a link or the position of the singularities of acurve. For example the Alexander polynomials of Zariski’s sextics are enough to show that they aretopologically distinct.

These invariants illustrate the several connections between the study of algebraic curves and knottheory: the characteristic varieties were first introduced by Hillman [Hil81] for links, then studiedby Arapura [Ara97] for Kahler manifolds, and first applied to algebraic curves by Libgober [Lib01].The Alexander polynomial, which is the first and most famous knot invariant, was introduced byAlexander [Ale28] and Fox [Fox62a] and developped for curves by Artal, Libgober, or Oka, amongmany others. For knots, it was studied from many different points of view: topological (as theorder of twisted homology of the complement, from Seifert forms,....), combinatorial (with coloringof diagrams or Skein relations), algebraic (as an invariant of the group or of the monodromy in thecase of fibred knots, via braid group representations),.... For curves, it can be described topologicallyin relation with cyclic branched coverings or algebraically, in terms of certain linear systems relatedto the singularities of the curve. Moreover, it coincides with the characteristic polynomial of themonodromy of the Milnor fiber.

Much more common objects are studied and applied in both settings, but there is one speciallyimportant: the braid and mapping class groups. Braid groups appeared historically with the worksof Hurwitz [Hur91] on ramified covering of surfaces and Magnus [Mag34]. Their beauty andwealth comes from the several ways to define them, as trajectories of particles, fundamental groupof configuration spaces (Fox-Neuwirth [FN62]), algebraic objects (Artin [Art25a]) or mapping classgroups. They were used in knot theory via a famous theorem of Alexander that expresses any linkas the closure of a braid. From that braid, it is possible to obtain invariant of the link, as theAlexander and Jones polynomial, among others.

Similarly, braids appear as a main tool in the understanding of the topology of curves vie itsbraid monodromy. This invariant, relatively to a given pencil of lines, provides complete informationabout the embedding of the curve (as shown by Kulikov-Teicher [KT00] and Carmona [Car03]). Itcan be understood as a formalization of the Zariski-van Kampen method to give a presentation ofthe fundamental group of a plane curve complement, as Chisini realized in the thirties. Much later,it was extensively used by Moishezon [Moi81]. Then Libgober proved that the homotopy type ofthe complement of an affine curve can be retreived from the presentation of the fundamental groupissued from braid monodromy, as in the case of links. For line arrangements (curves with componentsof degree one), the braid monodromy can be encoded in wiring diagrams [Arv92, CS97], that allowcombinatorial approaches of topological invariants similarly to knot diagrams.

The memoire is divided into four chapters. In the first, we briefly recall the twisted homology andintersection, and related invariants as the Alexander modules and the Franz-Reidemeister torsion. Inthe second, we present our construction of colored link signatures and derived necessary conditionsfor the existence of surfaces in smooth 4-manifolds. We also present a project, whose a part isstill in progress, on the (non)-additivity of the link signature under the splice operation. The third

6 CONTENTS

chapter is devoted to curve complements and twisted Alexander polynomials. We also construct anew topological invariant of line arrangements, closely related to the characteristic varieties. Thelast and fourth chapter concerns a functorial description of the abelian Reidemeister torsion of3-manifolds and links. We discuss some perspectives on the generalisation to 4-manifolds.

Finally, I should say that I started my research activity by reading the book A la recherche dela topologie perdue containing a set of papers collected and commented by L.Guillou and A.Marin.I have many souvenirs of my first talks on Rohlin’s signature theorem or on the proof of Ω3 = 0 byKneser method or even on Gordon’s proof of the G-signature theorem and the construction of theCasson-Gordon invariants. The mathematics a la Russe are always very inspiring for me.

1. BIBLIOGRAPHY OF THE AUTHOR 7

1. Bibliography of the author

[13] Alexander invarians of ribbon tangles and planar algebras, (with C.Damiani)2016, Submitted

[12] A functorial extension of the Magnus representation to the category of three-dimensional cobor-disms (with G.Massuyeau, J. Serrano)

2016, Submitted

[11] The signature of a splice (with A.Degtyarev, A.Lecuona)International Mathematics Research Notices 2016

[10] A topological invariant of line arrangements (with E.Artal, B.Guerville)To appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

[9] On complex line arrangements and their boundary manifolds (with B.Guerville, M.Marco)Mathematical Proceedings of the Cambridge Philosophical Society (2015) 159, pp 189-205

[8] A functorial extension of the abelian Reidemeister torsion of three-manifolds (with G.Massuyeau)L’Enseignement Mathmatiques 61:1/2 (2015) 161-210

[7] Alexander representation of tangles (with S.Bigelow, A.Cattabriga)Acta Vietnamica Mathematica (2015) Vol. 40 (2) pp 339-352

[6] Braids in Pau- An Introduction (with E.Artal)Annales Mathmatiques B.Pascal 18, 1-14 (2011)

[5] Twisted Alexander polynomial of plane algebraic curves (with J.I.Cogolludo)Journal of London Math. Society (2007) Vol. 76 Part 1, p105-121

[4] Generalized Seifert surfaces and signature of colored links (with D.Cimasoni)Transaction of the American Mathematical Society 360, (2008) p1223-1264

[3] On the Fox-Milnor theorem for the Alexander polynomial of linksInternational Mathematics Research Notices (2004) 2, p55-67

[2] On the slice genus of links (avec P.Gilmer)Algebraic and Geometric Topology (2003) 3, p905-920.

[1] Signatures of colored links with an application to real algebraic curvesJournal of Knot Theory and its Ramifications (2005) 14 , no. 7, p883-918

[1] collects the results of my PHD thesis. [2],. . . ,[5] were written during my postdoc years, and[6],. . . ,[13] during my position at the University of Pau.

8 CONTENTS

2. List of co-authors

E.Artal, Universidad de Zaragoza, Espagne

S.Bigelow, University of California, USA

A.Cattabriga, Universita di Bologna, Italie

J.I.Cogolludo, Universidad de Zaragoza, Espagne

D.Cimasoni, Universite de Geneve, Suisse

A.Degtyarev, Bilkent University, Turkey

P.M.Gilmer, Louisana State University, USA

A.Lecuona, Universite d’Aix-Marseille, France

M.Marco, Universidad de Zaragoza, Espagne

G.Massuyeau, Universite de Strasbourg, France

B.Guerville was PHD student, in cotutelle with E.Artal. He has a postdoc position in Tokyo.

J.Serrano is a PHD student, in cotutelle with E.Artal and G.Massuyeau.

C.Damiani is a PHD student, at the Universite de Caenunder the supervision of P.Bellingeri and E.Wagner.

3. STRUCTURE OF THE MEMOIRE 9

3. Structure of the memoire

Chapter 1 is devoted to general definitions on twisted homology and intersection, and Reide-meister torsion. Experts may skip it or use it as a reference for notations.

The results are collected in three different chapters, each having its own introduction and aSection with perpectives for the future.

Chapter 2: Links and surfaces in 4-manifolds

. New family of concordance invariants of colored links, derived from signatures of twisted inter-section of complement of surfaces. Calculation from a generalization of Seifert surfaces to coloredlinks, properties related to the characteristic varieties of the link.

Results by myself, and with D.Cimasoni. [1,4]

. Necessary conditions for the existence of smooth surfaces in B4, with prescribed Betti numbersand boundary a given colored link.

Results by myself, with D.Cimasoni and with P.Gilmer [1,2,3,4]

. A detailed study of the behavior of the signature of colored links under the splice operation.

Part of a current project with A.Degtyarev and A.Lecuona [11]

Chapter 3: Plane algebraic curves and line arrangements

. Twisted Alexander polynomial and Reidemeister torsion of curves, relation with local polynomials

Results with J.I.Cogolludo [5]

. A new topological invariant of line arrangements, some relations with characteristic varieties,computation with wiring diagrams

Results with E.Artal, B.Guerville and a part with M.Marco [9,10]

Chapter 4: TQFT models for the Alexander polynomial, and Reidemeister torsion

. Construction of a 2 + 1-Reidemeister functor

Results with G.Massueyau, with S.Bigelow and A.Cattabriga for tangles [7,8]

. Construction of a Magnus functor (extension of the representation)

Results with G.Massueyau and J.Serrano [12]

. Planar algebras and Alexander polynomials of 4-dimensional ribbon cobordisms

Results with C.Damiani [13]

CHAPTER 1

Preliminaries

This chapter is devoted to twisted homology and intersection, and Franz-Reidemeister torsion.We do not pretend that the framework is the most general as possible. Indeed, several variationswill be considered in this memoire, and the details will be given in the context. Our aim here isto briefly recall how interrelated are the characteristic varieties and the Alexander invariants of aCW-complex, with respect to the torsion, and, for manifolds, the role of the intersection pairings induality.

1. Twisted homology

We defined the twisted homology of a CW-complex. More details can be found in [Tur00]. Let

X be a finite CW-complex with π := π1(X). The universal covering p : X → X has a CW-structureinduced by X. The orientations of cells are chosen such that the restriction of p to each cell is

orientation preserving. The action of π by deck transformations on X induces an action on the

cellular chain groups C∗(X), which extends by linearity to an action of the group ring Z[π]. For aright Z[π]-module M , we consider the complex

C∗(X;M) := M ⊗Z[π] C∗(X;Z)

and its homology H∗(X;M) := H∗(C∗(X;M)). For a left Z[π]-module N , the twisted cohomology

H∗(X;N) is defined as the homology of HomZ[π](C∗(X;Z), N). Similar constructions apply to a

CW-pair (X,Y ), starting with (X, p−1(Y )).An important case arises as follows: γ : π → G is a group epimorphism and M = Z[G] be the

induced Z[π]-module. If XG → X is the G-covering defined by γ, then there is a chain Z[G]-linearisomorphism C∗(X;Z[G]) = C∗(XG;Z). The homology H∗(X;M) is sometimes denoted Hγ

∗ (X;M),or even simply Hγ

∗ (X), to emphasize the dependance on γ.Denote X → X the maximal abelian covering defined by π → H1(X) = π/[π;π]. The complex

C∗(X;Z) is a complex of Z[H1(X)]-modules.

Example 1.1. Suppose that H is free abelian with a basis t1, . . . , tµ. Let γ : π → H1(X)→ Hbe a group epimorphism and XH → X be the H-covering defined by γ. Then the following Z[H]-complexes are isomorphic

C∗(X;Z[H]) = Z[H]⊗Z[H1(X)] C∗(X;Z),

where Z[H] = Z[t±11 , . . . , t±−1

µ ] is the ring of Laurent polynomials. Note that the fraction field ofthis integral domain is denoted Q(H) = Q(t1, . . . , , tµ).

11

12 1. PRELIMINARIES

Example 1.2. Let ω : π → C∗ be a multiplicative torsion character. Denote C(ω) the field Cregarded as a Z[π]-module via ω. Suppose that ω factors as

ω : π → Gχ→ C∗,

where G is a finite abelian group and denote XG → X the covering induced by χ. There is anobvious canonical isomorphism between H∗(X;C(ω)) and the χ-equitypical summand

V χ∗ (G) :=

x ∈ H∗(XG;C)

∣∣ gx = χ(g)x for all g ∈ G,

In particular, H∗(X;C(ω)) does not depend on G as an intermediate group. Alternatively, C(ω) canbe regarded as a local system on X, and H∗(X;C(ω)) is the ordinary homology of X with coefficientin this local system. See also [Sak95].

2. Intersection forms

In this section we define the intersection pairing in homology with twisted coefficients. Let R bea field with involution r 7→ r. Let V be a left R-module together with a non-singular sesquilinearinner product (, ) : V × V → R. This means that for all v, w ∈ V and r ∈ R we have

(rv, w) = r(v, w), (v, rw) = r(v, w)

and (, ) induces an R-module isomorphism V ' HomR(V,R). An element A of GLR(V ) is unitaryif (Av,Aw) = (v, w) for all v, w ∈ V .

Let X be an oriented smooth manifold of dimension n. Consider a unitary representationα : π → GL(V ) inducing a right Z[π]-module structure on V . Let V ′ = V be equipped with theleft Z[π]-module structure given by gv = vα(g−1) for v ∈ V and g ∈ π. The following map is awell-defined left R-module homomorphism

HomZ[π](C∗(X), V ′) −→ HomR(V ⊗ C∗(X, R)

f 7→(w ⊗ c 7→ (f(c), w)

).

This gives rise to isomorphismsH∗(X;V ′)→ H∗(HomR(V⊗Z[π]C∗(X, R). By the univeral coefficienttheorem, the evaluation homomorphism now induces an isomorphism of left R-modules

H i(X;V ′) −→ HomR(Hi(X,V ), R),

where we equip H∗(·, V ) and H∗(·, V ) with the left R-module structures given on V . Combiningwith the Poincare duality isomorphism (cf [Wal99]) we get

Hi(X,V )→ Hi(X, ∂X, V ) ' Hn−i(X,V ′)→ HomR(Hn−i(X,V ), R),

and a R-sesquilinear pairing

Hi(X,V )×Hn−i(X,V ) −→ R.

If n = 2i, then the pairing ϕ is hermitian. This is the intersection pairing of X with twistedcoefficients V .

In this memoire, except for twisted Alexander polynomials considered in Chapter 3 Section 2(where more details are given), we mainly work with (abelian) one-dimensional representations andR can be either the fraction field Q(H) of a free abelian group H, or R = C.

3. CHARACTERISTIC VARIETIES AND ALEXANDER INVARIANTS 13

Example 2.1. Consider the context of Example 1.2, where α = ω is a 1-dimensional unitaryrepresentation with R = C and V has dimension 1 (a unitary character). Suppose that ω is torsion

and factors through ω : π → Gγ→ C∗, where G is a finite group. Then the induced covering XG is

an oriented rational homology manifold and we have a well-defined Hermitian intersection form

〈 · , · 〉 : H2(XG;C)⊗H2(XG;C)→ C.

Regard the homology groups H∗(XG;C) as C[G]-modules and consider the form

ϕ : H2(XG;C)⊗H2(XG;C)→ C[G], ϕ(x, y) :=∑g∈G〈x, gy〉g.

Since G is abelian, this form is sesquilinear, i.e., ϕ(g1x, g2y) = g1g−12 ϕ(x, y) for all g1, g2 ∈ G. This

induces a C-valued Hermitian form ϕχ on H2(X;C(ω)); explicitly, the latter is given by

ϕχ(x⊗ z1, y ⊗ z2) = z1z2

∑g∈G〈x, gy〉χ(g).

Recall that there is a canonical isomorphism between Hω∗ (X;C(ω)) and the χ-equitypical sum-

mand V χ∗ (G). The form ϕχ is |G|-times the restriction to V χ(G) of the ordinary intersection index

form 〈 · , · 〉. Now, if G is replaced with a larger group G′ G, the transfer map induces an isomor-phism V χ

∗ (G)→ V χ∗ (G′), multiplying the intersection index form by another positive factor [G′ : G].

It is worth noticing that the signature is preserved, and then depends only on ω.

3. Characteristic varieties and Alexander invariants

The characteristic varieties and the Alexander polynomials are interrelated invariants.Let X be a finite CW-complex, and let H be a free abelian group with basis t1, . . . , tµ. We fix

an epimorphism γ : π → H, and consider only characters ω : π → C∗ that factor through H. Thegroup of such characters can be identified with the complex torus (C∗)µ via ωi := ω(ti). In practice,in this memoire, the morphism γ will basically be fixed by a geometric construction (for curves orlinks, the classes of meridians provide a unique geometric basis of H1(X)). For ω ∈ (C∗)µ, let C(ω)be the field C viewed as a Z[π]-module. Note that if ω is torsion, ie if ωi are roots of unity, we arein the construction of Example 1.2.

Definition 3.1. The characteristic varieties of X(relative to γ) are defined as

Vr(X, γ) :=ω ∈ (C∗)µ

∣∣ dimH1(X;C(ω)) ≥ r, r ≥ 0.

The characteristic varieties are nested (Vr ⊃ Vr+1) algebraic varieties in (C∗)µ. They dependonly on the fundamental group π of X (and γ).

Let R be a unique factorization domain (basically a Euclidian ring), and N be a finitely generatedR-module. A presentation of N is an exact sequence of the form Rm → Rn → N → 0. The matrixof the homomorphism Rm → Rn with respect to the standard bases in Rm and Rn is a presentationmatrix of N . The kth-elementary ideal of N is the ideal Ek(N) generated by the (n− k)-minors ofa presentation matrix. Elementary ideals do not depend on the choice of the matrix, and dependonly on N . Recall that the greatest common divisor of an ideal E is the generator of the smallestprincipal ideal containing E. Let ∆k(N) = gcd(Ek(N)) ∈ R, be defined up to multiplication by aunit of R. The order of N is ∆(N) := ∆0(N).

Suppose now that R = Z[H] = Z[t±11 , . . . , t±1

µ ] is a Laurent polynomial ring.

14 1. PRELIMINARIES

Definition 3.2. The kth Alexander polynomial of (X, γ) is ∆k(X, γ) := ∆k(H1(X;H)) ∈ R.The Alexander polynomial of (X, γ) is the order of H1(X;H).

For I ∈ Z[H] , we denote VC(I) the ideal of I ⊗ C ⊂ C[H] in (C∗)µ. According to Lib-gober [Lib01], there are isomorphisms

Vk(X) \ 1 = VC(Ek−1(H1(X;ZH))) \ 1, k ≥ 1,

where 1 is the trivial character. These isomorphims are sometimes taken for the definition of Vr(X),which extends to any finitely generated ZH-module. In particular, it follows that the irreduciblecomponents of Vk(X) of codimension ≤ 1 constitute the zero locus of the (k − 1)th Alexanderpolynomial ∆k−1(X, γ).

4. Combinatorial torsion

We recall the definition and basic properties of the torsions of chain complexes. The reader isreferred to [Mil66] and [Tur01] for further details and references.

4.1. Definition of the torsion. Let F be a field. Given an F-vector space V of finite dimensionn ≥ 0, an n-tuple b = (b1, . . . , bn) of vectors in V and a basis c = (c1, . . . , cn) of V , we denote by[b/c] ∈ F the determinant of the matrix expressing b in the basis c. Two bases b and c are said tobe equivalent if [b/c] = 1.

Given a short exact sequence of F-vector spaces 0 → V ′ → V → V ′′ → 0 and some bases c′

and c′′ of V ′ and V ′′ respectively, we denote by c′c′′ the equivalence class of bases of V obtained byjuxtaposing (in this order) the image of c′ in V and a lift of c′′ to V .

By a finite F-chain complex of length m ≥ 1, we mean a chain complex C in the category offinite-dimensional F-vector spaces and we assume that C is concentrated in degrees 0, . . . ,m:

C =(Cm

∂m // Cm−1// · · · ∂1 // C0

).

A basis of C is a family c = (cm, . . . , c0) where ci is a basis of Ci for all i ∈ 0, . . . ,m. A homologicalbasis of C is a family h = (hm, . . . , h0) where hi is a basis of the i-th homology group Hi(C) for alli ∈ 0, . . . ,m. If we have choosen a basis bj of the space of j-dimensional boundaries Bj(C) :=Im ∂j+1 for all j ∈ 0, . . . ,m − 1, then a homological basis h of C induces an equivalence class ofbases of Ci for any i: specifically, we consider the basis (bihi)bi−1 of Ci obtained by juxtapositionin the following short exact sequences where we denote Zi(C) := Ker ∂i:

0 −→ Bi(C) −→ Zi(C) −→ Hi(C) −→ 0(4.1)

and 0 −→ Zi(C) −→ Ci∂i−→ Bi−1(C) −→ 0.(4.2)

Definition 4.1. The torsion of a finite F-chain complex C of length m, equipped with a basisc and a homological basis h, is the scalar

τ(C; c, h) :=

m∏i=0

[(bihi)bi−1/ci

](−1)i+1

∈ F \ 0.

It is easily checked that this definition does not depend on the choice of b0, . . . , bm and, when C isacyclic, we set τ(C; c) := τ(C; c,∅).

4. COMBINATORIAL TORSION 15

The next result relates, in a specific situation, the torsion to alternated product of orders ofmodules (the homology torsion). Suppose that R is a Noetherian factorisation domain (all its idealsare finitely generated).

Theorem 4.2. (Turaev) Let C∗ be a bsaed free chain complex of finite rank over R such thatrankHi(C∗) = 0 for all i. Let R be the fraction field of R. Then the based chain complex C∗ =R⊗R C∗ is acyclic and

τ(C) =∏i

∆0

(Hi(C∗)

)(−1)i+1

.

A generalisation of this result to the case rankHi(C∗) 6= 0 were obtained by Kirk and Livingston,see [KL99].

4.2. The Milnor torsion. The Milnor torsion of a CW-complex is a specialisation of theReidemeister-Franz torsion. Let X be a finite CW-complex. We consider a free abelian group Hand an epimorphism γ : π → H, see Example 1.1. If C∗(X;Q(H)) is acyclic, then the Milnor torsionof (X, γ) is

τ(X, γ) := τ(C∗(X;Q(H))

)∈ Q(H).

The Turaev torsion is a generalization of Milnor torsion to any abelian group. Because of thechoice of the orders, orientations, and lifts of the cells of X, the torsion τ(X, γ) is only defined up tomultiplication by an element of ±H. It is invariant under cellular subdivision. Using Whitehead’stheory of smooth triangulations, the torsion applied to a smooth triangulations produces an invariantof smooth 3 or 4-manifolds.

A direct consequence of Theorem 4.2, the Milnor torsion can be expressed in terms of ordersof the Z[H]-modules H∗(C(X;Z[H]). This gives especially an argument to show that the Milnortorsion is a homotopy invariant. A useful particular case is given in the following theorem.

Theorem 4.3. Let X be a finite connected 2-dimensional CW-complex with χ(X) = 0.If b1(X) =1, let t be the generator of the infinite cyclic group H = H1(X)/TorsH1(X). Then we have

τγ(X) =

∆(X, γ) (t− 1)−1 if b1(X) = 1∆(X, γ) if b1(X) ≥ 2.

4.3. Duality theorem. In this section, we recall the duality theorem for torsion, in our context.It is due to Franz and Milnor [Mil62] (see [KL99] for non-acyclic complexes and twisted coefficientsby unitary representations).

X is now a compact smooth 4-manifold, with boundary ∂X, possibly empty. We endow X withthe CW-decomposition induced by a triangulation. Since X is compact, the CW-complex is finite.

Let us fix the following convention for manifolds with boundary. We say that (Y, υ) is theboundary of (X, γ) if ∂X = Y as manifolds and the two following diagrams commute , where i∗ isinduced by the inclusion:

π1(Y )i∗ //

υ""

π1(X)

γ||||

H

16 1. PRELIMINARIES

Theorem 4.4. If Cγ∗ (X;Q(H)) is acyclic, then Cυ∗ (∂X;Q(H)) is also acyclic and the followingholds:

τ(∂X, υ) = τ(X, γ) · τ(X, γ).

The version of Theorem 4.4 for non-acyclic complexes, due to [KL99], involves determinant ofintersection forms.

CHAPTER 2

Surfaces in 4-manifolds and colored links

An oriented knot in S3 is slice if it is the boundary of a smooth disk embedded into the 4-ball B4.One may also think of the knot as the cross-section of a smooth 2-sphere S2 in R4 by a hyperplaneR3. Artin [Art25b] was the first to show that there are 2-spheres in R4 whose cross-sections arenon trivial knots, or even pairs of spheres whose cross section are non trivial links. Later, Fox andMilnor [FM66] used the Alexander polynomial to prove that there exist non slice knots (the trefoilfor example). This is related to concordance, where knots are concordant if they bound a smoothlyembedded cylinder S1 × [0; 1] in S3 × [0; 1]. A knot is then slice if and only if it is concordant tothe unknot. See the survey [Liv05]. Note that there exists locally flat versions of sliceness andconcordance; the invariants and obstruction involved in this chapter mainly occur in this category.

Levine [Lev69a, Lev69b] showed that a slice knot is algebraically slice, i.e. any Seifert formof a slice knot is metabolic. He also showed that the converse holds in high odd dimension, i.e. anyalgebraically slice knot is slice. This comes to show the importance of Seifert forms in the study ofconcordance; however the result is false in dimension 3: Casson and Gordon [CG78, CG86] provedthat certain two-bridges knots in S3 which are algebraically slice are not slice.

In the early sixties Trotter introduced a numerical knot invariant called the signature [Tro62],which was subsequently extended to links by Murasugi [Mur65]. This invariant was generalized toa function (defined via Seifert forms) on S1 ⊂ C by Levine and Tristram [Tri69, Lev69a], invariantby concordance on a dense subset of S1.

In the case of links, they are several generalizations of sliceness. Fox [Fox62b] defined slice linksin the strong sense as links concordant to the trivial link (or equivalently bound disjoint disks). TheTristram- Levine signature vanishes for slice links in the strong sense, on a dense subset. Moreover,the question of detecting the (non) existence of smooth disks in B4 can be extended to any compactoriented surface (without close components) with prescribed topology, whose boundary is a givenlink in S3. The Murasugi-Tristram inequality relates the values of signature to the Betti numbersof a bounding smooth surface in B4. This provides an obstruction, easily applicable.

Later on, the signature was reinterpreted in terms of coverings and intersection forms of 4-manifolds by Viro [Vir73, Vir09]. This point of view has opened many opportunities of generaliza-tions. Levine [Lev69a, Lev69b] also constructed signatures for links such that the linking numberof any two components is zero by applying the Atiyah-Singer invariant [AS68] to the finite cycliccoverings of the zero surgery manifold on the link. They are concordance invariants and vanish forslice links. The Casson-Gordon invariants are also obtained from dihedral coverings of the manifoldobtained by a zero surgery on the knot. See also [Liv02].

17

18 2. SURFACES IN 4-MANIFOLDS AND COLORED LINKS

If the surface is not connected, we can encode the repartition of its components along the linkwith a coloring (the components are endowed with some integer in 1, . . . , µ) 1. Note that ingeneral, under some trivial conditions on the linking numbers, such surfaces always exist but witharbitrary high genus. The problem can be formulated as finding estimations of the colored slice genusof a link, defined as the minimal genus of a smooth suface in B4, bounding the link with respectto the coloring. More generally, one may allow components of the surfaces to intersect together,along ordinary double points for example. In this chapter, we present new invariants and necessaryconditions for the existence of bounding surfaces in B4 which take the coloring into account, andthe eventual existence of singularities.

In the first section, we define the signature of a colored link in an integral homology sphere.If the homology sphere is S3, we obtain a several variables generalization of the Levine-Tristramsignature. Our definition follows Viro’s approach and the G-signature theorem. Then, we introducegeneralized Seifert forms (which were first considered by Cooper [Coo82] for 2-component links).They allow to calculate the signature (or to define it directly), to compute a presentation of theAlexander module of the colored link, and then to show that the signature is a locally constantmap whose jumping loci are determined by the characteristic varieties. In Section 2 we collect somenecessary conditions for the existence of smooth or immersed surfaces in B4 with prescribed topologyand boundary a given colored link. We put emphasis that working with links whose componentsmay have non-zero linking number (and then allow singular surfaces in both dimension 3 and 4) isone of the original point of the work presented in this chapter. For each of these results, they areexamples showing that the obstruction is stronger than the previous existent obstruction (see theauthor publications for the explicit examples). Section 3 is devoted to the study of the behaviorof the signature under the splice operation. We show that the signature is almost additive, witha correction term generically independent of the links. We interpret this correction term as thesignature of a generalized Hopf link and give a simple closed formula to compute it. In the the lastsection we present our current research on the subject, with a particular attention to the definitionof a new invariant of colored links, the slope, whose definition was first motivated by non-genericcases of the splice formula for the signature.

1. Signature of colored links

1.1. Historic on Levine-Tristram signatures. Let V be a Seifert matrix for a link L in S3.Then, A(t) = V − tV T is a presentation matrix of the Alexander Z[t±1]-module of L. In particular,the Alexander polynomial ∆L of L is given by the determinant of A(t). If ω 6= 1 is a unit moduluscomplex number, then H(ω) = (1−ω)A(ω) is a Hermitian matrix whose signature σL(ω) and nullityηL(ω) do not depend on the choice of V . This yields integral valued functions σL and ηL defined onS1 \ 1. In the case of ω = −1, this signature was first defined by Trotter [Tro62] and studied byMurasugi [Mur65]. The more general formulation is due to Levine [Lev69a] and Tristram [Tri69],and referred to as the Levine-Tristram signature.

The functions σL and ηL are easily seen to be locally constant on the complement in S1 \ 1of the roots of ∆L. Also, ηL is related to the first Betti number of the finite cyclic coverings of theexterior of L. Moreover, when restricted to roots of unity of prime order, the signature and nullity

1We proved in [1] that the existence of a smooth embedding of the same abstract surface in B4 depends really onhow the component are distributed on the link.

1. SIGNATURE OF COLORED LINKS 19

are concordance invariants. (The case of ω = −1 is due to Murasugi, and Tristram extended itto any ω of prime order.) Finally, the so-called Murasugi-Tristram inequality imposes a condition,expressed in terms of the values of σL and ηL, on the Betti numbers of a smooth oriented surface inB4 spanning L. This inequality implies in particular that if L is slice in the strong sense [Fox62b],then σL vanishes at roots of unity of prime order.

At that point, all the methods of demonstration were purely 3-dimensional. A new light was shedon this theory in the early seventies. Building on ideas of Rokhlin [Roh71], Viro [Vir73] was ableto interpret the Levine-Tristram signature as a 4-dimensional object. Indeed, he showed that for allrational values of ω, σL(ω) coincides with the signature of an intersection form related to a cycliccover of B4 branched along a Seifert surface for L pushed in the interior of B4. This 4-dimensionalapproach was used by Kauffman and Taylor [KT76] to obtain a short proof of the Murasugi-Tristraminequality, in the case ω = −1. They were also able to prove an inequality relating the values ofσL and ηL to the genus of a closed oriented smooth surface in S4 that intersects the standardlyembedded 3-sphere in L. In particular, if there exists such a surface of genus 0 (that is, according to[Fox62b], if L is slice in the ordinary sense), then σL(−1) = 0. This 4-dimensional interpretationwas used with great success by several authors [Gil81, GLM81, Lev69b, Lev94, Smo89]. Seealso [CG86, Gil82, GL92].

1.2. Construction. We will use the following notations.

Notations:

Tµ = S1 × · · · × S1 = (S1)µ,

Tµ∗ = (S1 \ 1)µ,T µ =

(ω1, . . . , ωµ) ∈ (S1)µ ⊂ Cµ

∣∣ ωj = exp(2πiθj), θj ∈ Q,

TµP = elements of T µ of prime power order .

1.2.1. Signatures of 4-manifolds. We start with recalling the definition and some properties ofthe twisted signature of a 4-manifold.

Let N be a compact smooth oriented 4-manifold with boundary and G a finite abelian group.Fix a covering NG → N , possibly ramified, with G the group of deck transformations. If the coveringis ramified, we assume that the ramification locus F is a union of smooth compact surfaces Fi ⊂ Nsuch that

(1) ∂Fi = Fi ∩ ∂N ;(2) each surface Fi is transversal to ∂N , and(3) distinct surfaces intersect transversally, at double points, and away from ∂N .

Items (1) and (2) above mean that each component Fi of F is a properly embedded surface. Forshort, a compact surface F ⊂ N satisfying all conditions (1)–(3) will be called properly immersed.

Under these assumptions, NG is an oriented rational homology manifold and have a well-definedHermitian intersection form. Consider χ : G → C∗ and denote the C-valued Hermitian form ϕχ

on Hχ2 (N,F ). See Chapter 1 Examples 1.2 and 2.1 for a similar construction. We will denote by

sign(N) the ordinary signature of the 4-manifold N , i.e., that of the form 〈 · , · 〉 on H2(N). Thetwisted signature, denoted by signχ(N,F ), is the signature of ϕχ.

1.2.2. Colored links. Let L be a µ-colored link in an integral homology sphere S. By Alexanderduality, the group H1(SrL) is generated by the meridians of the components of L. We shall denote


by mki the meridians of the components of the sublink Li of L of color i = 1, . . . , µ. Let H be the

free multiplicative group generated by t1, . . . , tµ. The coloring on L gives rise to a homomorphism

c : H1(S r L)→ H, mki 7→ ti, i = 1, . . . , µ.

We consider multiplicative characters H1(SrL)→ C∗ that respect the coloring, i.e., factor through c.They are determined by their values on the generators ti, and the group of such characters can beidentified with T µ. Through this identification, the character ω ∈ T µ assigns the meridians of thecomponents of the sublink Li to ωi. With a certain abuse of the language, we will shortly speakabout the character ω on L and say that ω assigns ωi to (each component of) Li.

The next proposition asserts that ω : H1(S r L) → C∗ extends to a finite order characterω : H1(N r F ) → C∗ (also denoted by the same letter ω), where N is a 4-manifold bounded by Sand F ⊂ N is a certain properly immersed surface.

Proposition 1.1. Let L be a µ-colored link in an integral homology sphere S. Then, there existsa 4-manifold N and an oriented properly immersed surface F = F1 ∪ . . . ∪ Fµ in N such that

• ∂N = S and ∂Fi = Li for i = 1, . . . , µ,• the group H1(N r F ) ' Zµ is freely generated by the meridians mi of Fi, and• one has [Fi, ∂Fi] = 0 in H2(N, ∂N).

As a consequence, any character ω ∈ T µ extends to a unique character

ω : H1(N r F )→ C∗, mi 7→ ωi.

Definition 1.2. A pair (N,F ) that verifies the conclusion of Proposition 1.1 is called a spanningpair for (S, L) and F is a spanning surface for L.

For short, as for characters on links, we will sometimes speak about the character ω on F andsay that ω assigns ωi to the component Fi.

Definition 1.3. The signature of a µ-colored link L ⊂ S is the map

σL : T µ −→ Zω 7−→ signω(N,F )− sign(N),

where N and F are as in Proposition 1.1.

The signature of a colored link is well defined, i.e., independent of the pair (N,F ) chosen tocompute it. This is a consequence of Novikov’s additivity and the G-signature theorem.

Remark 1.4. The signature signω(N,F ) is closely related to the intersection form defined onthe homology of N \ F with coefficients in C twisted by ω, in the sense of Chapter 1, Section 1 and2. This is discussed in Section 4.1.

The following remark should be clear from the definition of the signature of a colored link.

Remark 1.5. Let L be a µ-colored link in S, and let ω ∈ T µ be a vector such that ωi = 1.Then, the following equality holds: σL(. . . , 1, . . . ) = σ

L1∪...∪Li∪...∪Lµ(. . . , 1, . . . ).

Another important observation is the fact that the coloring of the link is essential: it is notenough to merely assign a value of a character to each component of the link. More precisely, wehave the following relation. The extra term is due to the perturbation of the union Fµ ∪ Fµ+1 oftwo components of the ramification locus into a single surface.


Proposition 1.6. Let L := L1 ∪ . . .∪Lµ+1 be a (µ+ 1)-colored link, and consider the µ-coloredlink L′ := L′1 ∪ . . .∪L′µ defined via L′i = Li for i < µ and L′µ = Lµ ∪Lµ+1. Then, for any characterω ∈ T µ, one has

σL′(ω) = σL(ω1, . . . , ωµ, ωµ)− `k(Lµ, Lµ+1).

In particular, Proposition 1.6 provides a relation between the restriction of the multivariatesignature of a colored link to the diagonal in T µ and the Levine–Tristram signature of the underlyingmonochrome link.

1.3. Generalized Seifert surfaces. In this section, links are in S3.Recall that a Seifert surface for a link is a connected compact oriented surface smoothly embed-

ded in S3 that has the link as its oriented boundary. The notion of C-complex, as introduced in[Coo82] and [Cim04], is a generalization of Seifert surfaces to colored links.

Definition 1.7. A C-complex for a µ-colored link L = L1∪ · · ·∪Lµ is a union S = S1∪ · · ·∪Sµof surfaces in S3 such that:

• for all i, Si is a Seifert surface for Li (possibly disconnected, but with no closed components);• for all i 6= j, Si ∩ Sj is either empty or a union of clasps (see Figure 1);• for all i, j, k pairwise distinct, Si ∩ Sj ∩ Sk is empty.

The existence of a C-complex for a colored link is fairly easy. In the case µ = 1, a C-complexfor L is nothing but a (possibly disconnected) Seifert surface for the link L. Let us now define thecorresponding generalization of the Seifert form.

A 1-cycle in a C-complex is called a loop if it is an oriented simple closed curve which behaves asillustrated in Figure 2 whenever it crosses a clasp. Clearly, there exists a collection of loops whosehomology classes form a basis of H1(S). For any loop x, we defined iε([x]) as the class of the 1-cycleobtained by pushing x in the εi-normal direction off Si for i = 1, . . . , µ. The fact that x is a loopensures that this can be done continuously along the clasp intersections. There are more intrinsicdefinition of iε, see Section 1.5.1 above, but maybe less intuitive. Finally, let

αε : H1(S)×H1(S)→ Z

be given by αε(x, y) = lk(iε(x), y), where lk denotes the linking number.Let L be a µ-colored link. Consider a C-complex S for L and the associated Seifert matrices

Aε with respect to some fixed basis of H1(S). Let A(t1, . . . , tµ) be the matrix with coefficients inZ[t1, . . . , tµ] defined by

A(t1, . . . , tµ) =∑ε

ε1 · · · εµ t1−ε1

21 · · · t

1−εµ2

µ Aε,

iSjS

Figure 1. A clasp intersection.


Figure 2. A loop crossing a clasp.

where the sum is on the 2µ possible sequences ε = (ε1, . . . , εµ) of ±1’s. For ω = (ω1, . . . , ωµ) inTµ = S1 × · · · × S1 ⊂ Cµ, set

H(ω) =

µ∏i=1

(1− ωi)A(ω1, . . . , ωµ).

Using the fact that A−ε = (Aε)T , one easily checks that H(ω) is a Hermitian matrix. Recall thatthe eigenvalues of such a matrix H are real. Its signature sign(H) is defined as the number ofpositive eigenvalues minus the number of negative eigenvalues. The nullity (H) is the number ofzero eigenvalues of H.

Definition 1.8. Let Tµ∗ be the open subset (S1 \ 1)µ of the µ-dimensional torus Tµ ⊂ Cµ.The signature and nullity of the µ-colored link L are the functions

σL, ηL : Tµ∗ −→ Z

given by σL(ω) = sign(H(ω)) and ηL(ω) = (H(ω)) + β0(S)− 1, where β0(S) denotes the number ofconnected components of S.

By Sylvester’s theorem, σL(ω) and ηL(ω) do not depend on the choice of a basis of H1(S).

Theorem 1.9. The signature σL and nullity ηL do not depend on the choice of the C-complexfor the colored link L. Hence, they are well-defined as isotopy invariants of the colored link L.

This theorem relies on the following lemma (see [Cim04] for the proof).

Lemma 1.10. Let S and S′ be C-complexes for isotopic colored links. Then, S and S′ can betransformed into each other by a finite number of the following operations and their inverses:

• ambient isotopy;• handle attachment on one surface;• addition of a ribbon intersection, followed by a ‘push along an arc’ through this intersection

(see Figure 3);• the transformation described in Figure 3.

Example 1.11. If µ = 1, then the colored link L is just a link. Furthermore, a C-complex Sfor L is nothing but a (possibly disconnected) Seifert surface for L. Finally, A− is a usual Seifertmatrix A, and A+ = AT . Hence, the corresponding Hermitian matrix is given by

H(ω) = (1− ω)(AT − ωA) = (1− ω)A+ (1− ω)AT .

So if µ = 1, the signature of L is the Levine-Tristram signature of the link L.


T2 T3

Figure 3. The transformations T2 and T3 in Lemma 1.10.

212 S SLL1− +

Example 1.12. Consider the 2-colored link L illustrated above. A C-complex S for L is alsogiven. We compute Aε = (−1) if ε1 = ε2, and Aε = (0) else. Hence,

H(ω1, ω2) = (1− ω1)(1− ω2)(−1− ω1ω2) = −2<((1− ω1)(1− ω2)).

So σL(ω1, ω2) is given by the sign of −<((1 − ω1)(1 − ω2)). Furthermore,ηL(ω1, ω2) = 1 if ω1ω2 = −1, and ηL(ω1, ω2) = 0 else. Let us draw thedomain T 2

∗ as a square. The value of the function σL can be represented asillustrated opposite. Note that σL and ηL are constant on the connectedcomponents of the complement of the zeroes of ∆L(t1, t2) = t1t2 + 1, theAlexander polynomial of L. We shall explain this fact in Section 1.5.

−1

1

1

0 0

Let us point out an interesting consequence of Proposition 1.6: it is possible to compute thesignature and nullity of a µ-colored link by considering any finer coloring of the same underlying link.In particular, all the signatures (corresponding to all the possible colorings) can be computed fromthe signature corresponding to a coloring with the maximal number of colors. Using This simplifiesgreatly the computations in many cases , as illustrated by the following (didactic) example.

Example 1.13. Let us try to compute the Levine-Tristram signature of the link L illustratedbelow. One possibility is to choose a Seifert surface for L and to compute the corresponding Seifertmatrix. On the other hand, consider a 3-colored link L′ obtained by coloring the components of Lwith three different colors. There is an obvious contractible C-complex for L′, so σL′ is identicallyzero. By Proposition prop:coloring, the Levine-Tristram signature of L is given by

σL(ω) = σL′(ω, ω, ω)− 2 = −2

1.4. Seifert and intersection forms. Let L be a µ-colored link in S3. Consider a connectedC-complex S ⊂ S3 for L. By pushing S in B4 we obtain a spanning surface for L, properly immersedin the sense of Section 1.2.1. Indeed, clasps become ordinary double points in B4. Hence any ω ∈ TµQinduces a character H1(B4 \ F )→ C∗ and a branched covering of B4.


Theorem 1.14. Let S be a C-complex for the µ-colored link L in S3. Let F be the surface withboundary L obtained by pushing S in B4. Then, for all ω ∈ T µ, the matrix H(ω) derived fromSeifert forms on S in S3 is a matrix for the intersection form of (B4, F ) twisted by ω, up to amultiple.

The proof of Theorem 1.14 consists in an explicit geometric description of the covering of B4,branched along F , induced by ω. We show that, up to a multiple, H(ω) is a matrix for thecorresponding twisted intersection form in some convenient basis.

Corollary 1.15. The signature of the colored link defined in Section 1.3 and the signaturedefined in Section 1.2 coincide on T µ.

1.5. Caracteristic varieties.1.5.1. Alexander modules. Let L be µ-colored link in S3 with exterior X := S3 \ T (L). Let H

be a free abelian group with basis t1, . . . , tµ. By Section 1.2.2, the coloring induces an epimorphismc : H1(X)→ H. See Chapter 1 Section 3 for notations on Alexander modules.

Definition 1.16. The Alexander module of the colored link L is the Z[t±11 , . . . , t±1

µ ]-moduleH1(X;Z[H]). The Alexander polynomial of L is ∆L := ∆0(X; c).

The aim of this section is to show how the covering of X induced by c can be constructed froma C-complex for L. This leads to a presentation of the Alexander module of L and generalizes acelebrated theorem of Seifert, which corresponds to the case µ = 1.

Consider a C-complex S = S1 ∪ · · · ∪Sµ for L such that each Si is connected and Si ∩Sj 6= ∅ forall i 6= j. (Such a C-complex exists by transformations T1 and T2 of Lemma 1.10.) For i = 1, . . . , µ,choose some interior point vi of Si \

⋃j 6=i Si ∩ Sj . Given a clasp in Si ∩ Sj with i < j, consider an

oriented edge in Si∪Sj joining vi and vj and passing through this single clasp as described in Figure

2. This leads to a collection of oriented edges e1ij , . . . , e

c(i,j)ij , where c(i, j) denotes the number of

clasps in Si∩Sj (that is: the number of connected components of Si∩Sj). Let Kij ⊂ Si∪Sj denotethe graph given by the union of these edges. Finally let Kµ be the complete graph with verticesvi1≤i≤µ and edges e1

ij1≤i<j≤µ.

Lemma 1.17. The homology of S = S1 ∪ · · · ∪ Sµ is equal to

H1(S) =⊕

1≤i≤µH1(Si) ⊕

⊕1≤i<j≤µ

H1(Kij) ⊕ H1(Kµ).

Furthermore, a basis of H1(Kij) is given by < βìj >1≤`≤c(i,j)−1, where βìj = eìj − e`+1ij . Finally, a

basis of H1(Kµ) is given by < γ1ij >2≤i<j≤µ, where γijk = e1ij − e1

ki + e1jk.

Let Ni = Si × [−1, 1] be a bicollar neighborhood of Si. Given a sign εi = ±1, let Sεii be the

translated surface Si×εi ⊂ Ni. Also, let Y be the complement of⋃µi=1 Ni in X. Given a sequence

ε = (ε1, . . . , εµ) of ±1’s, set

Sε =

µ⋃i=1

Sεii ∩ Y.


Since all the intersections are clasps, there is an obvious homotopy equivalence between S and Sε

inducing an isomorphism H1(S)→ H1(Sε). Then, the map iε : H1(S)→ H1(S3 \ S) defined in Sec-tion 1.3 can be described as the composition of this isomorphism with the inclusion homomorphismH1(Sε)→ H1(S3 \ S).

Theorem 1.18. Let L = L1∪· · ·∪Lµ be a colored link, and consider a C-complex S = S1∪· · ·∪Sµfor L such that each Si is connected and Si ∩ Sj 6= ∅ for all i 6= j. Let α : H1(S) ⊗Z Λµ →H1(S3 \ S)⊗Z Z[H] be the homomorphism of Λµ-modules given by

α =∑ε

ε1 · · · εµ tε1+1

21 · · · t

εµ+1

2µ iε,

where the sum is on all sequences ε = (ε1, . . . , εµ) of ±1’s. Then, the Alexander module of L admitsthe finite presentation

H ⊗Z Z[H]α−→ H1(S3 \ S)⊗Z Z[H] −→ H1(X;Z[H]) −→ 0,

where H =⊕

1≤i≤µH1(Si)⊕⊕

1≤i<j≤µH1(Kij)⊕⊕

1≤i<j<k≤µ Zγijk and α is given by

• α =∏n6=i(tn − 1)−1α on H1(Si) for 1 ≤ i ≤ µ;

• α =∏n6=i,j(tn − 1)−1α on H1(Kij) for 1 ≤ i < j ≤ µ;

• α(γijk) =∏n6=i,j,k(tn − 1)−1α(γijk) for 1 ≤ i < j < k ≤ µ.

For colored links with 1, 2 or 3 colors, this result provides a square presentation matrix ofthe Alexander module expressed in terms of the Seifert matrices Aε. More precisely, we have thefollowing corollaries.

Corollary 1.19 (Seifert [Sei35]). Let A be a Seifert matrix for the link L. Then, tA− AT isa presentation matrix of the Alexander module of L.

Similarly, we get the following result.

Corollary 1.20 (Cooper [Coo82]). Let S = S1 ∪ S2 be a C-complex for a colored link L =L1 ∪L2. Let A (resp. B) be a matrix of the form α−− (resp. α−+) with respect to a basis of H1(S)adapted to the decomposition H1(S) = H1(S1)⊕H1(S2)⊕H1(K12). Then, a presentation matrix ofthe Alexander module of L is given by

(t1t2A− t1B − t2BT +AT ) ·D,where D = (Dij) is the diagonal matrix given by

Dii =

(t2 − 1)−1 if 1 ≤ i ≤ β1(S1);

(t1 − 1)−1 if β1(S1) < i ≤ β1(S1) + β1(S2);

1 if β1(S1) + β1(S2) < i ≤ β1(S),

and β1(·) denotes the first Betti number.

The case µ = 3 is similar, but the complete statement is a little cumbersome. Instead, let usgive an example of such a computation.

Example 1.21. Consider the 3-colored link L given in Figure 1.21. A C-complex S for L isalso drawn. Clearly, H1(S) = Zγ. Furthermore, lk(γε, γ) = 1 if ε1 = ε2 = ε3, and all the otherlinking numbers are zero. Hence, a matrix of α = α is given by (t1t2t3 − 1), and H1(X;Z[H]) =Z[t±1

1 , t±12 , t±1

3 ]/(t1t2t3 − 1).


+

−γ−

L

L

1 2

3

SL

Figure 4. The 3-colored link of Example 1.21.

For colored links with µ ≥ 4 colors, the presentation of Theorem 1.18 has deficiency(µ−1

3

),

so the corresponding presentation matrix is not square. This is not a surprise. Indeed, Crowelland Strauss [CS69] proved that if an ordered link has µ ≥ 4 components and if ∆L 6= 0, then itsAlexander module does not admit any square presentation matrix. Their proof applies to coloredlinks as well. Therefore, it is not possible to get a presentation matrix of H1(X;Z[H]) using theSeifert matrices Aε if µ > 3. Nevertheless, it is possible to compute the Alexander invariants up tosome indeterminacy. More precisely, let Z[H]l denote the localization of the ring Z[H] with respectto the multiplicative system generated by ti − 11≤i≤µ.

Corollary 1.22. Let L be a µ-colored link. Consider a C-complex S for L such that Si isconnected for all i and Si ∩ Sj 6= ∅ for all i 6= j. Then the corresponding matrix

A(t1, . . . , tµ) =∑ε

ε1 · · · εµ t1−ε1

21 · · · t

1−εµ2

µ Aε

is a presentation matrix of the Z[H]l-module H1(X;Z[H])⊗Z[H]Z[H]l. In particular, for all k, thereare non-negative integers mi such that the following equality holds in Z[H]:

µ∏i=1

(1− ti)mi ∆k(L) = ∆k(A(t1, . . . , tµ)).

1.5.2. Piecewise continuity of the signature. Let L be a µ-colored link. The signature and nullityof L can be understood as functions

σL, ηL : Tµ∗ : −→ Z,

where Tµ∗ = (S1 \ 1)µ ⊂ Cµ. In this subsection, we state a ‘piecewise continuity’ result.Recall that X is the exterior of L, whose coloring induces c : H1(X) → H, where H is a free

abelian group with basis t1, . . . , tµ.

Definition 1.23. The characteristic varieties of the µ-colored link L are defined as

Vr(L) := Vr(X, c), r ≥ 0.

The characteristic varieties of L yield a finite sequence Tµ∗ = V0 ⊃ V1 ⊃ · · · ⊃ V`−1 ⊃ V` = ∅.Theorem 1.24. For all r, ηL is equal to r on Vr \ Vr+1, and σL is constant on the connected

components of Vr \ Vr+1.

2. CONCORDANCE AND SLICE GENUS 27

As a corollary, we obtain the following result that extends a well known property of the Levine-Tristram signature. Note that in the case lk(Li, Lj) = 0, it was already proved by the author in aprevious paper (see [1]).

Corollary 1.25. Let L be a µ-colored link. The map σL is constant and ηL vanishes on theconnected components of Tµ∗ \ ∆L = 0.

Before concluding this section, let us look back at the 2-colored link L given in Example 1.12.By Corollary 1.20, a presentation matrix of its Alexander module is given by (t1t2 + 1). Therefore,V0(L) = (t1t2 + 1) and Vr(L) = Λ2 for r ≥ 1, leading to

Σr =

T 2∗ if r ≤ 0;

(ω1, ω2) ∈ T 2∗ | ω1ω2 + 1 = 0 if r = 1;

∅ if r ≥ 2.

By Theorem 1.24, σL is constant on the connected components of T 2∗ \Σ1 and of Σ1. Furthermore,

ηL is equal to 0 on T 2∗ \ Σ1 and equal to 1 on Σ1. This coincides with the computations made in

Example 1.12.

2. Concordance and slice genus

2.1. Signatures. The properties of σL(ω) and ηL(ω) studied in this section do not hold for allω in Tµ∗ . We shall denote by TµP the dense subset of Tµ∗ constitued by the elements ω = (ω1, . . . , ωµ)which satisfy the following condition: there exists a prime p such that for all i, the order of ωi is apower of p.

We first state the invariance of the restriction of σL and ηL to TµP under (colored) concordance.

Definition 2.1. Two colored links L and L′ with ν components are said to be concordant ifthere exists a collection of smooth disjoint cylinders T1, . . . , Tν properly embedded in S3 × [0, 1],such that for all i, Ti is a concordance between components of L and L′ of the same color.

Theorem 2.2. For all ω ∈ TµP , σL(ω) and ηL(ω) are concordance invariants.

This result follows from the fact that the exterior of the concordance is a homology cobordism.The detailed proof can be found in [1, Theorem 4.15] for the case of colored links with lk(Li, Lj) = 0for all i 6= j. It obviously extends to the general case. Note that this theorem can also be viewed asa consequence of [GL92, Theorem 9].

Then, we show that the signature and nullity provide a lower bound for the genus of a surfacein B4 with boundary L.

Theorem 2.3. Suppose that F = F1 ∪ · · · ∪Fµ in B4 has boundary L. Set β1 =∑

i rankH1(Fi),and let c be the number of double points of F . Then, for all ω ∈ TµP ,

|σL(ω)|+ |ηL(ω)− µ+ 1| ≤ β1 + c.

The case c = 0 can be found in [1,Theorem 5.19]. The proof of this generalization is very similar.Finally the following deals with an analogous result concerning surfaces in S4 whose intersection

with a standardly embedded 3-sphere in equal to the colored link L.

Definition 2.4. Let S3 denote the standard embedding of the 3-sphere in S4. The slice genusgs(L) of a µ-colored link L is the minimal genus of a closed oriented smooth surface P = P1t· · ·tPµ ⊂S4 such that Pi ∩ S3 = Li for all i. A µ-colored link is said to be slice if its slice genus is zero.


Note that for such a surface to exist, we must have lk(Li, Lj) = 0 for all i 6= j. This definitionshould be understood as a unification of several well-known notions of ‘sliceness.’ Indeed, considerthe case µ = 1. A 1-colored link is slice if it is the cross-section of a smooth 2-sphere in S4, that is,using Fox’s terminology [Fox62b], if it is slice in the ordinary sense. On the other hand, considera ν-colored link with ν components. Such a colored link is slice if it is the cross-section of ν smoothdisjoint 2-spheres in S4. According to Fox, such a link is called slice in the strong sense.

The signature and nullity provide a lower bound for the slice genus of a colored link. Indeed, wehave the following generalization of [KT76, Theorem 3.13].

Theorem 2.5. For all ω in TµP ,

|σL(ω)| ≤ gs(L) + min(0, ηL(ω) + 1− µ).

2.2. Alexander polynomial and the Fox-Milnor Theorem. Fox and Milnor showed thatthe Alexander polynomial of a slice knot is of the form f(t) · f(t−1) for some integral polynomial f .The following generalisation to colored links is proved in [3].

Theorem 2.6. [3] Let L be a µ-colored link in S3. If L bounds a smooth compact orientedsurface F = F1 ∪ · · · ∪ Fµ in B4, with µ connected components such that ∂Fi = Li and χ(F ) = 1,

then there exists p ∈ Z[t1, . . . , tµ] such that, up to a unit in Z[t−11 , . . . , t−1

µ ]:

∆L(t1, . . . , tµ) ·µ∏j=1

(tj − 1)−χ(Fj) = p(t1, . . . , tµ) · p(t−11 , . . . , t−1

µ ).

The proof uses the interpretation of the Alexander polynomial as a Milnor torsion, see Chapter1 Section 2.6, and a duality theorem [Mil62].

2.3. Casson-Gordon invariants. This section is a short presentation of a work in collabora-tion with P.M.Gilmer [2]. The Casson-Gordon invariants of knots are a natural generalisation of theLevine-Tristram signature in terms of intersection forms related to dihedral coverings of the knotcomplement. They were used to show that the cobordism class of the Seifert form of a knot (thealgebraic concordance class) does not determine its concordance class. P.M.Gilmer [Gil82] usedCasson-Gordon invariants to give a new lower bound on the slice genus of a knot. We extend thisconstruction to links.

The linking form on a rational homology sphere M is a non-singular symetric pairing l : H1(M)×H1(M) → Q/Z defined by l(x, y) = (X.y)/n, where X is a 2-chain with boundary nx. Since l isnon-singular, the adjoint of l is an isomorphism H1(M) → H1(M)∗ = Hom(H1(M),C∗)) via the

map Q/Z → C∗ that sends ab to e

ab . It is metabolic with metabolizer H if there exists a subgroup

H of H1(M)∗ such that H⊥ = H.We define an invariant σ(L, χ) as a signature defect (through the G-signature theorem [AS68],

similarly to Section 1.3) applied to the pair (N2, χ), where N2 is the double covering of S3 branchedalong L and χ ∈ H1(N2)∗ has finite order. We considet the related nullity η(L, χ). If ∆L(−1) 6= 0,denote β the linking form of N2 and n the number of components of L. We show the followingresult:

Theorem 2.7. [2] Suppose that L bounds a connected oriented properly embedded surface F ofgenus g in B4 and that ∆L(−1) 6= 0. Then the form β can be decomposed as a direct sum β1 ⊕ β2

such that the following conditions hold

3. SIGNATURE OF A SPLICE 29

• β1 has an even presentation of rank 2g+ n− 1 and signature σL(−1), and β2 is metabolic.• There is a metabolizer for β2 such that for all characters χ of prime power order in this

metabolizer,|σ(M,χ) + σL(−1)| ≤ η(L, χ) + 4g + 3n− 2.

We study as an example a family of two component links, which have genus h Seifert surfaces.Using Theorem 2.7, we show that these links cannot bound a smoothly embedded surface in B4

with genus lower than h, while the Murasugi- Tristram inequality does not show this. In fact thereare some links with the same Seifert form that bound annuli in B4.

3. Signature of a splice

3.1. The set-up. Let L be a µ-colored link in an integral homology sphere S. The union of thecomponents of L given the same color i = 1, . . . , µ is denoted by Li. The signature of L is a Z-valuedfunction σL defined on the character torus T µ see Definition 1.3 for details. We let T 0 := 1 ∈ C.

Given a character ω ∈ T µ and a vector λ ∈ Zµ, we use the common notation ωλ :=∏µi=1 ω

λii .

Often, the components of L are split naturally into two groups, L = L′ ∪ L′′, on which thecoloring takes, respectively, µ′ and µ′′ values, µ′ + µ′′ = µ. In this case, we regard σL as a functionof two “vector” arguments (ω′, ω′′) ∈ T µ′ ×T µ′′ . We use this notation freely, hoping that each timeits precise meaning is clear from the context.

Clearly, in the definition of colored link, the precise set of colors is not very important; sometimes,we also admit the color 0. As a special case, we define a (1, µ)-colored link

K ∪ L = K ∪ L1 ∪ . . . ∪ Lµas a (1 + µ)-colored link in which K is the only component given the distinguished color 0. Here,we assume K connected; this component, considered distinguished, plays a special role in a numberof operations.

In the following definition, for a (1, µ∗)-colored link K∗ ∪ L∗ ⊂ S∗, ∗ = ′ or ′′, we denote byT ∗ ⊂ S∗ a small tubular neighborhood of K∗ disjoint from L∗ and let m∗, `∗ ⊂ ∂T ∗ be, respectively,its meridian and longitude. (The latter is well defined as S∗ is a homology sphere.)

Definition 3.1. Given two (1, µ∗)-colored links K∗ ∪ L∗ ⊂ S∗, ∗ = ′ or ′′, their splice is the(µ′ + µ′′)-colored link L′ ∪ L′′ in the integral homology sphere

S := (S′ r intT ′) ∪ϕ (S′′ r intT ′′),

where the gluing homeomorphism ϕ : ∂T ′ → ∂T ′′ takes m′ and `′ to `′′ and m′′, respectively.

3.2. The signature formula. The complex conjugation is denoted by η 7→ η. The samenotation applies to the elements of the character torus T µ, where we have ω = ω−1.

The linking number of two disjoint oriented circles K, L in an integral homology sphere S isdenoted by `kS(K,L), with S omitted whenever understood. For a (1, µ)-colored link K ∪L, we alsodefine the linking vector `k(K,L) = (λ1, . . . , λµ) ∈ Zµ, where λi := `k(K,Li).

The index of a real number x is defined via ind(x) := bxc − b−xc ∈ Z. The Log-functionLog : T 1 → [0, 1) sends exp(2πit) to t ∈ [0, 1). This function extends to Log : T µ → [0, µ) viaLogω =

∑µi=1 Logωi; in other words, we specialize each argument to the interval [0, 1) and add the

arguments as real numbers (rather than elements of T 1) afterwards. For any integral vector λ ∈ Zµ,µ ≥ 0, we define the defect function

δλ : T µ −→ Z


−1

1

δ(1,1)(ω)

0

δ(1,2)(ω)

−2

2

0 −1

1

−1

0

1

0

0

0

0 0

δ(2,3)(ω)

10

0

42

−2

−4

3

1

−1−3

1

−10

2

−2

0 −1

Figure 5. The values of three defect functions for ω ∈ T 2. The defect is constanton the shaded regions and on the interior of the segments dividing the squares. Thevalues of the defect in the extremal cases, ω1 = 1 or ω2 = 1, are given by the numberson the left and bottom of the squares respectively.

ω 7−→ ind(∑µ

i=1 λi Logωi)−∑µ

i=1 λi ind(Logωi).

For short, if λi = 1 for all i, we simply denote the defect δ, and omit the subscript. The reader isreferred to Figure 5 for a few examples of the defect function on T 2.

The following statement is the principal result of the paper.

Theorem 3.2. For ∗ = ′ or ′′, consider a (1, µ∗)-colored link K∗ ∪ L∗ ⊂ S∗, and let L ⊂ S bethe splice of the two links. For characters ω∗ ∈ T µ∗, introduce the notation

λ∗ := `k(K∗, L∗) ∈ Zµ∗, υ∗ := (ω∗)λ

∗ ∈ T 1.

Then, assuming that (υ′, υ′′) 6= (1, 1), one has

σL(ω′, ω′′) = σK′∪L′(υ′′, ω′) + σK′′∪L′′(υ

′, ω′′) + δλ′(ω′)δλ′′(ω

′′).

Remark 3.3. Eisenbud and Neumann [EN85, Theorem 5.2] showed that the Alexander poly-nomial is multiplicative under the splice. For a µ-colored link L, we denote ∆L(t1, . . . , tµ) the

Alexander polynomial of L. Similar to Theorem 3.2, let t∗ =∏µ∗

i=1(t∗i )λ∗i . One has

∆L(t′1, . . . , t′µ′ , t

′′1, . . . , t

′′µ′′) = ∆K′∪L′(t

′′, t′1, . . . , t′µ′) ·∆K′′∪L′′(t

′, t′′1, . . . , t′′µ′′),

unless µ′ = 0 (ie. L′ = K ′ is a knot) and λ′′ = 0, in which case

∆L(t′′1, . . . , t′′µ′′) = ∆L′′\K′′(t

′′1, . . . , t

′′µ′′).

Note that this formula were refined by Cimasoni [Cim05] for the Conway potential function. More-over, in relation with the signature of a colored link, one may consider the nullity, related to therank of the twisted first homology of the link complement. This nullity is also additive under thesplice operation, in the suitable sense. Detailed statements will be found in a paper in preparation.

Example 3.4. Consider two copies K ′ ∪ L′ and K ′′ ∪ L′′ of the (1,1)-colored generalized Hopflink H1,2, see Section 3.3, where K ′ and K ′′ are the single components. Then, L = L′ ∪ L′′ = H2,2

is a (1,1)-colored link, and for ω ∈ T 1 r ±1, we show by using C-complexes that

σL(ω, ω) = σK′∪L′(ω2, ω) + σK′′∪L′′(ω

2, ω) + δ(2)(ω)δ(2)(ω) = 0 + 0 + δ(2)(ω)δ(2)(ω).

This illustrates trivially that a defect appears.


K ′ L′ L′′1 L′′2

K ′′

L+ − + −

+

+ −

+

−

Figure 6. The leftmost link is the (2,4)-torus link, depicted as the boundary of aC-complex with rank 1 first homology. In the middle, the (4,2)-cable over the unknotwith the core retained, bounding a rank 2 C-complex. The last diagram is the spliceof the two preceding ones along K ′ and K ′′. It represents the (3,6)-torus link.

Example 3.5. For the reader convenience we add the following example. Notice the use of theformula in Theorem 3.2 when ωi = 1 (cf. Remark 1.5). Let K ′ ∪ L′ be the (2,4)-torus link andK ′′∪L′′ be the (4,2)-cable over the unknot with the core retained (cf. Section 3.5). Then, the spliceof these two links along the components K ′ and K ′′ is the (3,6)-torus link, which we shall denote L.

In the notation of Theorem 3.2, we have λ′ = 2 and λ′′ = (1, 1). For the C-complexes boundedby these three links one can take those depicted in Figure 6. To simplify the resulting Hermitianmatrices H, we re-denote by t0, t1, . . . their arguments (in the order listed) and, for an index set I,introduce the shortcut πI := 1 +

∏i∈I(−ti). Then

HK′∪L′(ξ′, ω′) = −π0π1π01,

HK′′∪L′′(ξ′′, ω′′1 , ω

′′2) = π0π1π2

(−π0π12 t1t2π0

π0 −π012

),

HL′∪L′′(ω′, ω′′1 , ω

′′2) = π0π1π2

−π0π12 t1t2π0 0 0π0 −π012 t0t2π1 t0π2

0 π1 −π1π02 −t0π1π2

0 t1π2 π1π2 −π2π01

,

so that, up to units and factors of the form πi, i = 0, 1, . . ., the Alexander polynomials are

∆K′∪L′ = π01, ∆K′′∪L′′ = t0t21t

22 − 1, ∆L′∪L′′ = π012(t0t1t2 + 1)2.

The computation of the signature of these matrices is straightforward: on the respective open tori,they are the piecewise constant functions given by the following tables:

Log ξ′ + Logω′ 1/2 3/2σK′∪L′(ξ

′, ω′) 1 0 −1 0 1

Log ξ′′ + 2 Logω′′ 1 2 3 4σK′′∪L′′(ξ

′′, ω′′) 2 1 0 −1 −2 −1 0 1 2

Logω′ + Logω′′ 1/2 1 2 5/2σL′∪L′′(ω

′, ω′′) 4 2 0 −1 −2 −1 0 2 4


Note, however, that L′ is the unknot and L′′ is homeomorphic to K ′ ∪ L′; hence,

σK′∪L′(1, ω′) = 0, σK′′∪L′′(1, ω

′′) = σK′∪L′(ω′′1 , ω

′′2).

Now, it is immediate that the identity

σL(ω′, ω′′1 , ω′′2) = σK′∪L′(ω

′′1ω′′2 , ω

′) + σK′′∪L′′(ω′2, ω′′1 , ω

′′2) + δ(2)(ω

′)δ(1,1)(ω′′1 , ω

′′2)

given by Theorem 3.2 holds whenever ω′2 6= 1 or ω′′1ω′′2 6= 1. (It suffices to compare the values at

all triples of 8-th roots of unity.) If ω′2 = ω′′1ω′′2 = 1, we obtain an extra discrepancy of 1; this

phenomenon is explained in Section 4.1.

As an immediate consequence of Theorem 3.2, we see that the Levine–Tristram signature of asplice cannot be expressed in terms of the Levine–Tristram signature of its summands: in general,the multivariate extension is required.

Corollary 3.6. Let L be the splice of (1, 1)-colored links K ′ ∪ L′ and K ′′ ∪ L′′, and denoteλ′ = `k(K ′, L′) and λ′′ = `k(K ′′, L′′). Consider L as a 1-colored link. Then, for a character ξ ∈ T 1

such that ξg.c.d.(λ′,λ′′) 6= 1, one has

σL(ξ) = σK′∪L′(ξλ′′ , ξ) + σK′′∪L′′(ξ

λ′ , ξ)− λ′λ′′ + δλ′(ξ)δλ′′(ξ),

where σL(ξ) is the Levine–Tristram signature of L.

3.3. The generalized Hopf link. A generalized Hopf link is the link Hm,n ⊂ S3 obtainedfrom the ordinary positive Hopf link H1,1 = V ∪ U by replacing its components V and U with,respectively, m and n parallel copies. This link is naturally (m + n)-colored; its signature, whichplays a special role in the paper is given by Theorem 3.7 below. Observe the similarity to thecorrection term in Theorem 3.2; a posteriori, Theorem 3.7 can be interpreted as a special case ofTheorem 3.2, using the identity σH1,n ≡ 0 (which is easily proved independently) and the fact thatHm,n is the splice of H1,m and H1,n. However, the Hopf links and their signatures are used essentiallyin the proof of Theorem 3.2.

Theorem 3.7. For any character (v, u) ∈ T m × T n, one has σHm,n(v, u) = δ(v)δ(u).

Certainly, Theorem 3.7 computes as well the signature of a generalized Hopf link equipped withan arbitrary coloring and orientation of components. First, one can recolor the link by assigning aseparate color to each component (cf. Proposition 1.6 below). Then, one can reverse the orientationof each negative component Li; obviously, this operation corresponds to the substitution ωi 7→ ωi.For example, the orientation of the original link can be described in terms of a pair of vectors, viz.the linking vector ν ∈ ±1m of the V -part of Hm,n with the U -component of the original Hopflink H1,1 and the linking vector λ ∈ ±1n of the U -part with the V -component. Then, assumingthat any two linked components of Hm,n are given distinct colors, we have

(3.1) σHm,n(v, u) = δν(v)δλ(u).

For future references, we state a few simple properties of the defect function δ and, hence, ofthe signature σHm,n . All proofs are immediate.

Lemma 3.8. The defect function δ : T µ → Z has the following properties:

(1) δ(1) = 0; δ ≡ 0 if µ = 0 or 1;(2) δ(ω) = −δ(ω) for all ω ∈ T µ;(3) δ is preserved by the coordinatewise action of the symmetric group Sµ;


(4) δ commutes with the coordinate embeddings T µ → T µ+1, ω 7→ (ω, 1);(5) δ commutes with the embeddings T µ → T µ+2, ω 7→ (ω, η, η) for any η ∈ T 1.

3.4. Satellite knots. As was first observed in [EN85], the splice operation generalizes manyclassical link constructions: connected sum, disjoint union and satellites among others.

Our first application is Litherland’s formula for the Levine–Tristram signature of a satellite knot.Recall that an embedding of a solid torus in S3 into another solid torus in another copy of S3 is

called faithful if the image of a canonical longitude of the first solid torus is a canonical longitude ofthe second one. Let V be an unknotted solid torus in S3, and let k be a knot in the interior of V ,with algebraic winding number q, i.e., [k] is q times the class of the core in H1(V ). Given any knotK ⊂ S3, the satellite knot K∗ is defined as the image f(k) under a faithful embedding f : V → S3

sending the core of V to K.The isotopy class K∗ depends of course on the embedding f (and even its concordance class,

see [Lit84]). Nevertheless, its Levine–Tristram signature is determined by the signatures of theconstituent knots and the winding number:

Theorem 3.9 (cf. [Lit79, Theorem 2]). In the notation above, the Levine–Tristram signaturesof k, K and K∗ are related via

σK∗(ω) = σK(ωq) + σk(ω), ω ∈ T 1.

3.5. Iterated torus links. Our next application is another special case of Theorem 3.2, whichprovides an inductive formula for the signatures of iterated torus links. In particular, this classof links contains the algebraic ones, i.e., the links of isolated singularities of complex curves in C2.Note that partial results on the equivariant signatures of the monodromy were obtained by Neumann[Neu87].

Iterated torus links are obtained from an unknot by a sequence of cabling operations (and maybe,reversing the orientation of some of the components). In order to define the cabling operations (wefollow the exposition in [EN85]), consider two coprime integers p and q (in particular, if one of themis 0, the other is ±1), a positive integer d, a (1, µ′)-colored link K ′ ∪ L′ ⊂ S3, and a small tubularneighbourhood T ′ of K ′ disjoint from L′. Let m, l be the meridian and longitude of K ′, and K ′(p, q)be the oriented simple closed curve in ∂T ′ homologous to pl + qm. More generally, let dK ′(p, q) bethe disjoint union of d parallel copies of K ′(p, q) in ∂T ′. We say that the link L = L′∪dK ′(p, q)−K ′(resp. L = L′∪dK ′(p, q)) is obtained from K ′∪L′ by a (dp, dq)-cabling with the core removed (resp.retained).

Let H1,1 = V ∪ U be the ordinary Hopf link. The link V ∪ dU(p, q) can be regarded as either(1, d)-colored or (1, 1)-colored. We denote the corresponding multivariate and bivariate signaturefunctions by τdp,dq and τdp,dq, respectively. Note that, by Proposition 1.6 below,

τdp,dq(v, u) = τdp,dq(v, u, . . . , u)− 12d(d− 1)pq.

In the case of core-removing, the link L obtained by the cabling is nothing but the splice of K ′ ∪L′and V ∪dU(p, q). (Similarly, in the core-retaining case, L is the splice of K ′∪L′ and V ∪U∪dU(p, q).)Hence, the following statement is an immediate consequence of Theorem 3.2.

Theorem 3.10. Let L be obtained from a (1, µ′)-colored link K ′ ∪ L′ by a (dp, dq)-cabling with

the core removed. For a character ω := (ω′, ω′′) ∈ T µ′ × T d, let

λ′ := `k(K ′, L′), λ′′ := (p, . . . , p) ∈ Zd, and υ∗ := (ω∗)λ∗, ∗ = ′ or ′′.


Then, assuming that (υ′, υ′′) 6= (1, 1), one has

σL(ω) = σK′∪L′(υ′′, ω′) + τdp,dq(υ

′, ω′′) + δλ′(ω′)δλ′′(ω

′′).

With the evident modifications, this last corollary can be adapted to give a formula for a (dp, dq)-cabling with the core retained.

The Levine–Tristram signature of the torus link U(p, q) (which coincides with τp,q(1, ζ) in ournotation) was computed by Hirzebruch. Unfortunately, we do not know any more general statement.

4. Perspectives

4.1. Slopes of colored links. The assumption (υ′, υ′′) 6= (1, 1) in Theorem 3.2 on the (non)-additivity of the signature under the splice operation is essential. If υ′ = υ′′ = 1, the expression forthe signature acquires an extra correction term, which can be proved to take values in [[−2, 2]]. Inmany cases, this term can be computed algorithmically, and simple examples show that typically itdoes not vanish.

Example 4.1. Consider two copies of the Whitehead link K ′ ∪ L′ and K ′′ ∪ L′′. If ω = eiπ/3,then σL(ω, ω) = −1, but σK′∪L′(1, ω) + σK′′∪L′′(1, ω) + δ(1) = 0 and there is a non-zero extra term.

The general computation of this extra term is the first motivation for this common project withA.Degtyarev and A.Lecuona. This appears to be a new concordance invariant of colored links, whosedependance on the fundamental group is not completly understood.

4.1.1. Extension of the signature of colored links to non-torsion characters. Our first aim is todefine the signature function of µ-colored links at any (non-rational) character in Tµ∗ = (S1 \ 1)µ.Note that Seifert forms allow such a (3-dimensional) construction for links in S3, but a 4-dimensionalapproach is needed for proving several properties.

Recall that a spanning pair (N,F ) for (S, L) consists in a 4-manifold N with boundary S, anda properly immersed surface F = F1 ∪ . . . ∪ Fµ in N such that ∂Fi = Fi ∩ ∂N = Li for i = 1, . . . , µ.

Conjecture 4.2. Let (N,F ) be a spanning pair of (S, L) with H1(N) = 0 and [Fi, ∂Fi] = 0 forall i. Set WF = N \ T (F ). For all ω ∈ Tµ∗ , the signature defect is a colored isotopy invariant of L

σL(ω) = signω(WF )− sign(WF ),

where signω(WF ) is the signature of the intersection form on H2(WF ;C(ω)).

The proof of the conjecture roughly follows two steps. First, the Atiyah-Patodi-Singer [APS75]theorem states that the signature defect is an invariant of the boundary ∂WF (and ω). Next, surgeryand cobordism arguments show that σL(ω) is indeed independant of the spanning surface F . Thenthe conclusion of Theorem 3.2 on additivity of signatures would still hold without the assumptionthat the characters must be rational (but unitary).

4.1.2. Slopes and η-functions. Typically, given a link L ⊂ S, we denote by TubL a small opentubular neighborhood of L. Let X := S \TubL and X := S \Tub(K ∪L). For a component C ⊂ L,we denote by ∂CX the intersection of ∂X with the closure of TubC. A (1, µ)-colored link is a(1 + µ)-colored link

K ∪ L = K ∪ L1 ∪ . . . ∪ Lµ ⊂ S

4. PERSPECTIVES 35

in which the knot K is the only component (considered distinguished) given the distinguishedcolor 0. We use the notation `k(K,L) ∈ Zµ for the linking coefficient vector (λ1, . . . , λµ), whereλi = `k(K,Li), i = 1, . . . , µ.

Consider a (1, µ)-colored link K ∪ L ⊂ S. A character ω ⊂ (C∗)µ of π is said admissible ifit restricts to the trivial character on H1(∂KX). Equivalently, ω regarded as a homomorphismH1(X) → C∗ is admissible if ω([K]) = 1. The variety of admissible characters is denoted byA(K/L) ⊂ (C∗)µ; it is the zero set of the polynomial ωλ−1, where λ := `k(K,L) and we abbreviate

ωλ :=∏i ω

λii . In other words, if λ = 0, then A(K/L) = (C∗)µ; otherwise, leting N := gcdλ

and ν := λ/N , the irreducible over Q components of A(K/L) are the zero sets of the cyclotomicpolynomials Φd(ω

ν), d | N . Since all varieties are defined over Z, for each component A ⊂ A(K/L)and each r ≥ 0, the complement A \ Vr(L) is either empty or dense in A.

We mainly consider the variety A(K/L) := A(K/L) ∩ (C∗ \ 1)µ of non-vanishing admissiblecharacters. Let ω ∈ A(K/L). Since ω is non-vanishing, we have H∗(∂LX;C(ω)) = 0 and, since ωis also admissible,

H1(∂X;C(ω)) = H1(∂KX;C(ω)) = H1(∂KX;C).

The latter isomorphism is canonical up to a multiplicative constant. Denote the inclusion i : ∂KX →∂X, and consider the subspace

Z(ω) := Ker i∗ : H1(∂KX;C(ω))→ H1(X;C(ω)).

Let m, l be the meridian and a preferred longitude of K.

Definition 4.3. Let ω ∈ A(K/L) and assume that dimZ(ω) = 1, ie, Z(ω) is generated by asingle vector am+ bl for some [a : b] ∈ CP 1(C). Then, the slope of K ∪ L at ω is the quotient

(K/L)(ω) := −ab∈ C ∪∞.

The definitions of Z(ω) and (K/L)(ω) extend to a character ω with ωi = 1 by patching thesublink Li. Thus, Z and K/L are defined on the whole variety A(K/L) of admissible characters.

By Poincare duality, if ω is unitary, we can prove that dimZ(ω) = 1 and (K/L)(ω) is real(possibly infinite).

Problem 4.4. Determine for which ω in Tµ∗ , the slope (K/L)(ω) is invariant by concordance.

It can be proved that (K/L)(ω) is a concordance invariant for all ω such that ∆L(ω) 6= 0. Thesituation of other values is not fully determined.

We show that the slope can be viewed as a generalisation of the Kojima-Yamasaki’s η-functionintroduced for 2-component links with linking number 0 [KY79] (the definition looks very different).Cochran showed that, up to a change of variable, the η-function is the generating function for asequence βi of invariants [Coc85] related to Sato-Levine [Sat84] invariants of successive derivativesof the link. They are the only known link invariants that vanish for boundary links. More recently,these Cochran invariants have been interpreted in terms of intersections of twisted Withney towersby Conant, Schneiderman and Teichner, see [CST16]. Note also that Cochran gave a family ofconcordance classes distinguished by these βi but whose Milnor’s µ-invariants and Murasugi 2-eightscoincide.


4.1.3. Rationality and Conway potential functions. Here, we state two theorems describing theslope as a rational function. For a µ-colored link L, we denote ∇L(t1, . . . , tµ) its Conway potentialfunction.

Given a complex number z = r exp(πiθ) 6= 0, we normalize its argument so that 0 ≤ θ < 1 andlet√z :=

√r exp(πiθ/2). For a character ω := (ω1, . . . , ωµ), we let

√ω := (

√ω1, . . . ,

√ωµ).

Theorem 4.5. Denote ∇′ = ∂

∂t∇K∪L. Then, for ω ∈ A(K/L), one has

∗(K/L)(ω) =

∇′(1,√ω)

2∇L(√ω), if ∇L(

√ω) 6= 0,

∞, if ∇L(√ω) = 0 and ∇′(1,√ω) 6= 0.

In particular, in both cases above the slope is well defined.

Remark 4.6. The mysterious polynomial 12∇′ in the statement of Theorem 4.5 can be under-

stood as follows: if ω is admissible, we have ∇K∪L(1,√ω) = 0, ie, ∇K∪L(t,

√ω) = (t− t−1)R(t) for

a certain Laurent polynomial R ∈ C[t±1], and we merely substitute t = 1 to the residual factor R.

Note that Theorem 4.5 is inconclusive if ∇L(√ω) = ∇′(1,√ω) = 0: just as in the freshman

calculus, the indeterminate form 0/0 should be resolved by other means.Recall that for k ≥ 0, Vk(L) are the chatacter varieties of L, related to the coloring.

Theorem 4.7. Pick a component A ⊂ A(K/L) and let r be the minimal integer such that∆L,r|A 6= 0, ie, A \ Vr+1(L) is dense in A. Denote by R the coordinate ring of A and fix anormalization of ∆L,r. Then, either

• there exists a unique polynomial ∆A ∈ R such that

(K/L)(ω) =∆A(ω)

∆L,r(ω)

holds for each character ω ∈ A \ Vr+1(L), or• the slope (K/L)(ω) =∞ is well defined and infinite at each character ω in a certain dense

Zariski open subset of A.

Case 4.7 cannot occur if r = 0, ie, if ∆L|A 6= 0, cf. Theorem 4.5.

Remark 4.8. The slope (K/L)(ω) at a character ω ∈ A∩Vr+1(L) does not need to be given bythe rational function in Theorem 4.7, even if the latter admits an analytic continuation through ω.

4.1.4. Slopes and signatures. Recall that the formula of Theorem 3.2 that relates the signatureof a splice link in terms of the signatures of its splice components is not available if (υ′, υ′′) =(1, 1). In this case both characters, ω′ and ω′′, are admissible and there is a well defined slopefor the corresponding colored links. An extra correction term appears, that can be expressed as acombination of the signs of the slopes of the links in the splice. The following theorem gives theprecise statement.

Theorem 4.9. For ∗ = ′ or ′′, consider a (1, µ∗)-colored link K∗ ∪ L∗ ⊂ S∗, and let L ⊂ S bethe splice of the two links. With the above introduced notation, if (υ′, υ′′) = (1, 1), then

σL(ω′, ω′′) = σL′(ω′) + σL′′(ω

′′) + δλ′(ω′)δλ′′(ω

′′)

4. PERSPECTIVES 37

+ sg(K ′′/L′′)(ω′′)− sg

(1

(K ′′/L′′)(ω′′)− (K ′/L′)(ω′)

).

The sign conventions in the preceding theorem are as follows.

sg(x) =

0 if x ∈ 0,±∞,∞−∞1 if x > 0−1 if x < 0

Note that in spite of the asymmetric appearance of the formula in Theorem 4.9 with respect to therole played by the slopes of K∗ ∪L∗, it is indeed unchanged if the roles of K

′ ∪L′ and K′′ ∪L′′ are

switched.Finally, in [4, Theorem 5.1] we present and generalize an idea of J. Conway leading to a purely

combinatorial computation of the signature for many colored links. If two colored links are related bya certain local move (surgery on a crossing), then the signature of them differ by a term dependingon the signs of their Conway polynomial. However, this formula is not available as soon as thispolynomial vanishes. We show in this current project that a general formula holds, involving theslopes of the links.

4.2. Slice links. This project is in collaboration with A.Conway (PHD Student at the Univer-site de Geneve) and A.Lecuona. Our aim is to extend the notion of sliceness from knots to (colored)links. We would like to have three different approaches: study the ribbon case, follow Levine in hisoriginal proof of the metabolic Seifert pairing and develop a 4-dimensional point of view.

Let L be a µ-colored oriented link in S3 and S be a connected C-complex S for L. Define thematrices AεS and A(t1, ..., tµ) as in [5] (see Section 1.3). Consider the matrix, hermitian for the

involution induced by ti → t−1i :

A(t1, ..., tµ) =

µ∏i=1

(1− t−1i )A(t1, ..., tµ).

Recall that, by Corrolary 1.22, one has up to factors 1− ti (and units)

∆L(t1, . . . , tµ) = detA(t1, ..., tµ).

In all this section, we suppose that ∆L 6= 0, in particular A(t1, ..., tµ) is non-singular.

4.2.1. Ribbon links. A ribbon surface is an immersed surface in S3 whose singularities are self-intersections along arcs such that the preimage of each arc consists of two arcs, one completely inthe interior and the other one having its two endpoints on the boundary. An oriented link boundingsuch a surface with Euler characteristic 1 in S3 is called a ribbon link. If the link is µ-colored, werequire that the surface has µ-components with respect to the coloring. Note that in the case ofknots the famous slice-ribbon conjecture states that being slice is equivalent to being ribbon.

The matrices Aε play the role of the Seifert matrix in the case of knots and by metabolic wemean here that there is a subspace of half dimension 2 on which the matrices Aε vanish. We showthat for ribbon links the matrices Aε can also be chosen to be metabolic.

Proposition 4.10. Let L be a ribbon µ-colored link in S3. Then, there exists a C-complex forL, built canonically from R, for which all the matrices Aε are metabolic.

2A: It is a priori not clear that the dimension of H1(S) is even.


Note that the subspace on which the matrices Aε vanish is common to all Aε. This implies thatif L is ribbon, then there exists f ∈ Z[t±1

1 , . . . , t±1µ ] such that, up to a unit and factors 1− ti,

∆L = f(t1, . . . , tµ)f(t−11 , . . . , t−1

µ ).

This last result was already observed by Orevkov, see [Ore02].

4.2.2. Witt classes. Let S = Z[t±11 , . . . , t±1

µ ] be the Laurent polynomial ring. The set M(S) ofisomorphism classes of non-singular hermitian forms on R-modules is a monoid under the directsum. A hermitian form is metabolic if it contains a Lagrangian. The Witt group of hermitian formsW (S) is the quotient of M(S) by the submonoid of metabolic forms.

Let L be a µ-colored link in S3 and F be a spanning surface for L in B4. Let WF be theexterior of F in B4 and H be a free abelian group with basis t1, . . . , tµ. Consider the isomorphismχ : H1(WF ) → H that sends the oriented meridian of Fi to ti. Let ϕtF be the intersection form onH2(WF ;Q(H)). Since ∆L 6= 0, one shows that the pairing ϕtF is non-singular.

Conjecture 4.11. Suppose that L bounds a smooth surface F with µ-components in B4 (withpossibly double-points between the components). The Witt class

wL = w(ϕtF ) ∈W (Q(H)))

is an isotopy invariant of L.

The conjecture can be proved by using arguments of Cha [Cha10] for µ ≤ 4 but we do not knowto prove the result in general. Due to this difficulty, the following results are proved only for µ ≤ 4.

Theorem 4.12. Suppose that µ ≤ 4. Let L be a µ-colored link with ∆L 6= 0. Suppose that Lbounds F in B4 smoothly embedded such that χ(F ) = 1. Then wL = 0 ∈W (Q(H))).

In particular ∆L is of the form f(t1, . . . , tµ)f(t−11 , . . . , t−1

µ ).

The last part of the statement is the Fox-Milnor theorem for colored links, proved by the author[3], see Section 2.2. Here we obtain the result by proving that ∆L is the discriminant of the Wittclass wL.

This theorem can in fact be reformulated with hypothesis involving only the topology of Fandof its embedding, avoiding the hypothesis ∆L 6= 0. Let ∆F be the Alexander polynomial of itscomplement in B4. We have

Theorem 4.13. Let L be a µ-colored link with µ ≤ 4. . Suppose that L bounds F in B4

smoothly embedded such that χ(F ) = 1 and ∆F 6= 0. Then wL = 0 ∈ W (Q(H))) and ∆L is of theform f(t1, . . . , tµ)f(t−1

1 , . . . , t−1µ ).

4.2.3. Levine’s proof. We prove

Theorem 4.14. Let L be a µ-colored link with µ ≤ 4. , and C be a C-complex for L. LetA = A(t1, . . . , tµ) be the matrix associated to C. If ∆L 6= 0 and L bounds F in B4 smoothlyembedded with χ(F ) = 1, then

w(A) = 0.

Our proof is purely 4-dimensional, and we would like to have an analogue of Levine’s argumentfor the colored link case.

4. PERSPECTIVES 39

4.3. Colored slice genus and Alexander polynomial. This project is in collaboration withP.Feller (Max Plank). The starting point is the following result of P.Feller.

For every oriented knot K in S3, we denote gtop4 (K) the topolgical slice genus of K, ie the

minimal genus of a locally flat oriented spanning surfaces for K in B4. Let ∆K be the Alexanderpolynomial of K.

Theorem 4.15. [Fel16] For every knot K, 2gtop4 (K) ≤ deg (∆K).

This result is a consequence (and a generalisation) of the famous Freedman’s disc theorem statingthat knots with trivial Alexander polynomial bound a locally flat disc in B4. Theorem 4.15 can beviewed as a contrasts to the well-known and elementary fact deg (∆K) ≤ 2g(K), where g(K) is the3-dimensional genus of K.

Conjecture 4.16. Let L be a µ-colored link in S3. Let F = F1 ∪ · · · ∪ Fµ be a spanning locallyflat surface of L in B4. Let c be the number of doubles points of F . Then, if ∇L is the Conwaypotential function of L, one has

2∑i

g(Fi) + 2(c− 1) ≤∑i

degti (∇L).

If L is a knot, Conjecture 4.16 specializes to Theorem 4.15. Note that since we consider degrees,Conway functions are more natural than Alexander polynomials. We first restrict to the case oftwo-component links and µ = 2. We show the following

Theorem 4.17. Let S = S1 ∪ S2 be a C-complex for L = L1 ∪ L2 with γ claps. Then,

degt1 (∇L) + degt2 (∇L) ≤ 2(g(S1) + g(S2)) + 2(γ − 1).

We conjecture that the natural generalisation of Theorem 4.17 holds for an arbitrary numberof colors. We also plan to use the following analogue of Freedman’s theorem for double points inlocally flat surfaces in B4:

Theorem 4.18. [Dav06] A two-component link with Alexander polynomial 1 bounds two disjointlocally-flat disks with an ordinary double point.

CHAPTER 3

Plane algebraic curves and line arrangements

The history starts with moduli space of plane curves and the stratification induced by thesingularities of the discriminant, whose study were initiated by Zariski [Zar29]. For short, a curvewill be a singular algebraic curve in the complex projective plane CP 2. The topology of a curveC is defined as the homeomorphism type of the pair (CP 2, C). It provides a natural invariant ofconnected components of equisingular curves.

Two curves C and C′ have the same combinatorics if there exist tubular neighborhoods T (C) andT (C′) and a homeomorphism ϕ : T (C) → T (C′) such that ϕ(C) = C′. By Waldhausen’s theory ofgraphed manifolds [Wad94], this is equivalent to have irreducible components with the same degrees,(local) singularities with the same topological type, and the same description of their branches. Oneeasily shows that the topology of a curve determines its combinatorics but Zariski [Zar31] provedthe existence of sextics with the same combinatorics and different embeddings. This illustrates thatcombinatorial strata may not be irreducible and that the position of singularities has an effect onthe topology of the curve. Other examples were given later by Artal [Art94] and Oka [Oka92].

The main example of Zariski is a family of irreducible sextics with six cusps with the specialproperty to lye in the same conic. By a procedure found with Van Kampen, using an auxiliary pencilof lines, he showed that the fundamental group of their complement is Z/2Z ? Z/3Z. Moreover, heobserved the existence of other sextics with six cusps (not lying in a conic) whose complementhave abelian fundamental group. This comes to show that the group is sensitive to the position ofsingularities. In general a presentation can be obtained with Zariski-Van Kampen’s approach, bycomputing the so-called braid monodromy first and then using the latter to write down the relations.The braid monodromy was developped after Zariski and Van Kampen, in a modern way, by Cheniot[Che73], Chisini and Moishezon [Moi81]. Note that the Hurwitz equivalence class is a deformationinvariant in itself and was successfully used by Artal-Carmona-Cogolludo-Tokunaga [ACCT01] todistinguish curves with isomorphic fundamental groups.

The Alexander polynomial of a curve depends on the local type of its singularities: Libgober[Lib82] showed that it divides the product of the local polynomials. This result was sharpenedby Degtyarev [Deg94], which described the type of singular points that may contribute to theglobal polynomial. In this memoire, we present a new geometric proof of this fact and show thatthe quotient is related to the determinant of a twisted intersection form of the curve complement.Inspired by the strong progress in knot theory, we define twisted Alexander polynomials of curvesand show a similar relation with local poynomials, for unitary representations. See Section 1 fora short review on braid monodromy and the homotopy type of a curve complement [Lib86] andSection 2 for the definition of twisted polynomials, a statement of this result and examples.

41

42 3. PLANE ALGEBRAIC CURVES AND LINE ARRANGEMENTS

Line arrangements are finite collections of complex lines in the projective plane that is, curveswhose irreducible components are all of degree one. The case of line arrangements is quite motivatingsince lines are non-singular and two lines intersect at a single point: the combinatorics of an arrange-ment can easily be encoded in the incidence graph. The example of Rybnikov [Ryb98, Ryb11]illustrates that even in this case, the fundamental group is sensitive to the position of singularities.Note that for real arrangements it is not known if the characteristic varieties depends only on thecombinatorics. Moreover, a theorem of Suciu [Suc11] states essentially that, unless the arrangementis a pencil, the multivariable Alexander polynomial of an arrangement is trivial. One may insteadwork with one-variable polynomials (the Oka polynomials) relative to the choice of a morphismπ1 → Z. However, an accurate choice might be difficult to make.

In Section 3, we introduce a new topological invariant of line arrangements, derived from acareful study of the boundary manifold and the peripheral structure. We explain how this invariantcan be computed from the wiring diagram, introduced by Arvola [Arv92] (see also [CS97]), andhow it provides a crucial ingredient to compute the depth of certain resonnant characters (ie somepoints in the characteristic varieties, corresponding to characters that are not fully ramified). Noteeven if several examples illustrate that it is sensitive to the position of singularities (see [GB16]),it is still unknown if the invariant depends only on the fundamental group.

The last Section 4 is devoted to perspectives on new constructions of topological invariants ofcurves or arrangements.

1. Braid monodromy and the homotopy type

In this section, we shortly review how Libgober described the homotopy type of a curve com-plement in terms of the braid monodromy relative to a generic projection.

Consider a linear projection π : C2 → C, i.e. such that:

(1) there are no vertical asymptotes,(2) the fibers are transversal to C except for a finite number of them which are either simple

tangents to a point of C or lines through a singular point of C transversal to its tangentcone.

Let P be the (finite) set of critical values of π. The braid monodromy of C is the homomorphism

ϑ : π1(C \ P) −→ Bd,

where Bd denotes the braid group, viewed as the mapping class group of a generic fiber relative toC, that is, (π−1(p), π−1(p) ∩ C) with p ∈ C \ P. Note that accurate packages have been developedto compute braid monodromies of curves with equations over the rationals by Bessis [Bes05] andCarmona [Car03].

We fix a basis αii=1,...,n of π1(C \ P) such that:

(1) αi = Ai · si · A−1 where Ai is a path joining p and pi ∈ P, si is the boundary of a smalldisc Di around pi and Ai = Ai \Di

(2) α1 · . . . · αn is homotopic to the boundary of a big disc containing P.

Also fix a basis γ1, . . . , γd of the (free) fundamental group π1(π−1(p) \ C). Note that each π−1(Ai)produces the collapse of mi points of π−1(p)∩C to a point on C, say Pi, where mi is nothing but themultiplicity of intersection of the line π−1(p) and the curve C at Pi. We will denote the meridians

2. ALEXANDER POLYNOMIALS 43

of such points by γi,1, ..., γi,mi . The action of ϑ(αi) on γi,ki , ki = 1, ...,mi can be decomposed via Aiand si as follows

ϑ(αi)(γi,ki) = σi(ωi,kiγi,kiω−1i,ki

)

where σi only depends on the local type of the singularity of the projection and γi,ki = ωi,kiγi,kiω−1i,ki

is homotopic to a meridian of C on S3i \ C (a small 3-dimensional sphere centered at Pi).

In the following theorem, the presentation of π1(X) is due to [Zar29, VK33]. Libgober [Lit84]used this presentation to describe the homotopy type of X.

Theorem 1.1. The two-dimensional complex associated with the following presentation of π1(X):

〈 γ1, . . . , γd | σi(γi,ki) = γi,ki , i = 1, . . . , n, ki = 1, . . . ,mi − 1 〉

has the homotopy type of X.

2. Alexander polynomials

In knot theory, a strategy to study problems that the Alexander polynomial is not strong enoughto solve is to consider non-Abelian invariants, twisted by a linear representation of the fundamentalgroup -see [Kit96, Lin01, Wad94], for instance. For mutation and concordance questions, Kirkand Livingston[KL99] developed their properties in the general context of CW-complexes. In thissection, we use their results to establish the relationship between the Alexander polynomial of aplane algebraic curve, twisted by a unitary representation, and the product of the local ones.

Going back to knot theory, Milnor showed that the Alexander polynomial essentially coincideswith the Franz-Reidemeister torsion of the link complement, see Chapter 1 Section 4.2. Turaevfurther developed this construction, which provided new proofs for several classical results. Thefirst version of twisted Alexander polynomial for knots was due to Lin. Wada generalized it as aninvariant of finitely presentable groups endowed with a representation, in terms of Fox calculus. ThenKitano showed that it coincides with a torsion of the knot complement, in the acyclic case. Kirkand Livingston extended his construction to the non-acyclic case, and any finite CW-complex. FromLibgober’s theorem on the homotopy type of the complement of the affine curve (see Section 1) therelationship between the twisted Alexander polynomial and the related torsion can be established,similar to those in knot theory. Morevoer, a ad-hoc duality theorem (see Chapter 1 Section 4.3)relates in this context the torsion of a 4-manifold to the determinant of the twisted intersection formand the torsion of its boundary. If we consider the complement of a tubular neighbrhood of a curve,the torsion of the boundary manifold coincides with the product of local torsion and the dualitytheorem gives the relation with the torsion of the curve complement, in terms of intersection forms.

It is worth noticing that in general, the difficulty to work with twisted Alexander polynomialcomes from the choice of a ’good’ representation. We plan to establish in the future methodsto construct or choose such representations from the combinatorics of the curve. In particular,we expect that to use twisted Alexander polynomials in order to distinguish Zariski pairs of linearrangements, the clever choice of representations could be deduced from the incidence graph (atleast in the particular case of representations in symmetric groups).

We end the Section with an example of nodal degenerations, first presented by Artal-Cogolludo-Tokunaga. For such degenerations, the changes in the fundamental group are very subtle to observe;this illustrates the strengh of twisted Alexander polynomials.


2.1. Twisted chain complexes. In this section, X is a finite CW-complex, with π = π1(X,x)for x ∈ X. Let us fix an epimorphism

ε : π −→ Z.Consider now an F-vector space V of finite dimension and a representation

ρ : π −→ GL(V ).

If X → X denotes the universal covering, the cellular chain complex C∗(X;F) is an F[π]-modulegenerated by the lifts of the cells of X. Consider the F[π]-module F[t±1]⊗F V , where the action isinduced by ε⊗ ρ, as follows:

(p⊗ v) · α = (pε(α))⊗ (ρ(α)v), α ∈ π.Let the chain complex of (X, ε, ρ) be defined as the complex of F[t±1]-modules:

Cε,ρ∗ (X,F[t±1]) = (F[t±1]⊗ V )⊗F[π] C∗(X;F).

It is a free based complex, where a basis is given by the elements of the form 1⊗ ei⊗ ck, where eiis a basis of V and ck is a basis of the F[π]-module C∗(X;F), obtained by lifting cells of X.

Remark 2.1. The construction can be easily extended to any epimorphism ε : π → Zµ for someµ, and chain complexes over F[t±1

1 , . . . , t±1µ ]. The advantage of working with the one-variable PID

F[t±1] is mainly computational.

A geometrical interpretation of Cε,ρ∗ (X;F[t±1]) was given in [KL99]. We briefly recall theirpoint of view. Consider X∞ the infinite cyclic covering induced by ε. For π = Ker ε = π1(X∞), therepresentation ρ restricts to

ρ : π −→ GL(V ).

The chain complex

Cρ∗ (X∞;V ) = V ⊗F[π] C∗(X)

can be considered as a complex of F[t±1]-modules, F[t±1] is a trivial F[π]-module, as follows:

tn · (v ⊗ c) = vγ−n ⊗ γncwhere γ ∈ π verifies ε(γ) = t. In [KL99, Theorem 2.1] it is shown that Cρ∗ (X∞;V ) and Cε,ρ∗ (X,F[t±1])are isomorphic as F[t±1]-modules.

2.2. Curves. Let C be an algebraic curve in C2 with r irreducible components. Let us denoteby X the complement in B4 of an open tubular neighborhood of C, for a sufficiently large ballB4 ⊂ C2. The group H1(X) = Zr is generated by the homology classes ν` of the meridians γ` of C`for ` = 1, . . . , r. Let q1, . . . , qr 6= 0 be integers with gcd(q1, . . . , qr) = 1. Consider

ε : H1(X) −→ Zwith ε(ν`) = q`. Let

ρ : π1(X) −→ GL(V ),

be a fixed representation where V is a finite dimensional F-vector space.

Definition 2.2. The torsion of (C, ε, ρ) is τε,ρ(C) = τε,ρ(X). Similarly, Alexander polynomialscan be defined as ∆i

ε,ρ(C) = ∆iε,ρ(X). As in the case of links, we will denote ∆1

ε,ρ(C)/∆0ε,ρ(C) simply

by ∆ε,ρ(C).


Proposition 2.3. For any algebraic curve C in C2, one has

τε,ρ(C) = ∆ε,ρ(C).In particular, if ρ is the trivial representation and qi = 1, then

(t− 1)τε,ρ(C) = ∆C .

In fact, the second statement holds for any epimorphism ε and τε(C) coincides with the cor-responding θ-polynomial introduced by Oka [Oka02]. Note that the complex Cε,ρ∗ (X;F(t)) is notacyclic in general. In fact even the Euler characteristic is not zero.

In some cases, one can assure that ∆ε,ρ(C) is actually a polynomial.

Proposition 2.4. If C is not irreducible and Hε,ρ1 (X;F[t±1]) is torsion, then ∆ε,ρ(C) is a poly-

nomial.

2.3. Local polynomials and divisibility. Suppose that we are given (C, ε, ρ). Let S31 , . . . , S

3s

be sufficiently small 3-spheres around the singular points P1, ..., Ps of C. Denote by Lk = C∩S3k the

link of the singularity at Pk, and by Ek be the link exterior. Also choose a base point Qi ∈ S3k \ Lk

and denote by πk = π1(S3k \ Lk;Qi) the local fundamental groups at Pk. The inclusion maps

ik : πk → π1(X) and (ε, ρ) induce morphisms

εk : πk −→ Zand ρk : πk −→ GL(V ),

for any k = 1, ..., s.

Definition 2.5. For all k = 1, . . . , s, let the local torsions be defined as

τk = τεk,ρk(Lk).

Analogously, by Chapter 1 Theorem 4.3, we define ∆k = ∆εk,ρk(Lk) = τk. Also, there is a localtorsion at infinity defined as

τ∞ = τε∞,ρ∞(L∞),

where L∞ = C ∩ ∂B4 is the intersection of the curve with the boundary of the big ball consideredat the beginning of section 2.2.

Note that an explicit description of the maps πk → π can be obtained via the braid monodromyof a generic projection of the curve, see Section 1. Note that, via the lifting of the paths Ak, themeridians γk,j can also be seen as meridians of C on S3

k based at Qk ∈ S3k \Lk. The local fundamental

group πk can be presented as

πk = 〈 γk,j | σk(γk,j) = γk,j , j = 1, ...,mk − 1 〉and ρk(γk,j) = ρ(ωk,jγk,jω

−1k,j ), for any j = 1, ...,mk − 1. A similar description of π∞, ε∞, and ρ∞

can be given analogously using ϑ(α1 · ... · αn) (as defined in 2) instead of the local braids.Let ϕε,ρ(C) be the intersection form of (X, ε, ρ), on Hε,ρ

2 (X;F[t±1]).

Theorem 2.6. [5] Let (C, ε, ρ) be unitary and suppose that the local representations ρk areacyclic. Then( r∏

`=1

det(Id−ρ(ν`)tq`)s`−χ(C`)

)·

∏k=1,...,s,∞

∆k = ∆ε,ρ(C) ·∆ε,ρ(C) · detϕε,ρ(C),


where ν` is the homology class of a meridian around the irreducible component C` and s` = #Sing(C)∩C`.

In the classical case, we obtain

Corollary 2.7. If ϕt(C) is the intersection form with twisted coefficients in Q[t±1], then

(t− 1)1−χ(C)∏

k=1,...,s,∞∆Lk = ∆2

C · detϕt(C).

2.4. Nodal degenerations. The following example illustrates how twisted Alexander polyno-mials can be sensitive to nodes, something that classical Alexander polynomials are not.

We say a curve D is of type I if D is an irreducible plane curve of degree d such that D has anordinary (d − 2)-ple point at P . Let L1 and L2 be lines through P such that either Li is tangentto a smooth point Pi ∈ D or Li passes through a double point Pi 6= P of type A2r. Let us denoteC = L1 + L2 + D. Assume that D has only nodes as singular points apart from P . Accordingto [ABCT06, Theorem 1], D is rational if and only if there exists a dihedral cover D2n ramifiedalong 2(L1 + L2) + nD for any n ≥ 3. In fact, according to [ABCT06, Corollary 2] this impliesthat D is rational if and only if the fundamental group of CP 2 \ (L1 ∪ L2 ∪ D) admits Z2 ∗ Z2 as aquotient. Moreover (see [ABCT06, Proposition 6.1]), there exist nodal degenerations Dt → D0 toa rational D0 of type I using (non-rational) curves Dt (t > 0) of type I with Abelian fundamentalgroups. A presentation for the fundamental group of C0 = L1 + L2 +D0 is given as follows:

G(C0) =〈 `, x1, x2 | [x1, x2] = 1, `−1x1` = x2, `−1x2` = x1 〉.

Considering ε the usual morphism ε(ν`) = 1, and

ρ(`) =

(1 00 −1

),

ρ(x1) =

(−1 01 −1

),

ρ(x2) =

(−1 0−1 −1

)one obtains

(2.1) ∆ε,ρ(C0) = t+ 1.

Note that ρ(G(C0)) ∼= Z2 ∗ Z2.The three non-nodal singularities of Ct are P, P1, P2 and they lie on the lines L1 and L2. Hence,

maybe by performing projective transformations, we can assume that P, P1, P2 and L1 and L2

are fixed throughout the degeneration. This implies that the classical Alexander polynomial ∆Ct ofCt is the same for all t ≥ 0 (since they have the same adjunction ideals, see for instance [Lib83,Theorem 5.1]). Since G(Ct) is Abelian, this implies that ∆C1 = ∆C0 = 1. Formula (2.1) shows thatC0 has a non-trivial twisted Alexander polynomial.

3. LINE ARRANGEMENTS 47

3. Line arrangements

In the eighties, Orlik and Solomon [OS80] showed that the cohomology ring of the complement ofan arrangement is determined by this incidence graph. This is not true for the deformation classes, asit was shown for MacLane combinatorics [BLVS+99a, Mac36]. Rybnikov [Ryb98, ACCAM07]constructed a pair of complex line arrangements with the same combinatorics but whose fundamen-tal groups are not isomorphic. This illustrates that the combinatorics of an arrangement does notdetermine its topology in general, and that even in this case, the fundamental group of the com-plement is a strong invariant. Other examples were exhibited in [ACCAM05] by using the braidmonodromy. As in the general case of curves, the group provides a difficult invariant to handle effec-tively and the derived Alexander modules and characteristic varieties were extensively studied (seefor example [CS99a, CS99b]. These invariants can be computed from the fundamental group, butthe task can be endless for present computers for most arrangements. Moreover, it is still unknownwhether characteristic varieties or Alexander polynomial are combinatorially determined (thoughthere are some partial results). Using a method by Ligboger [Lib01] it is possible to compute mostirreducible components of characteristic varieties (only some isolated points may fail to be found).A method to compute these extra components can be found in [AB14].

The boundary manifold BA, defined as the common boundary of a regular neighbourhood of Aand its exterior EA, is a graph 3-manifold whose structure is determined by the combinatorics ofA. See [CS08, CS06]. In a first work [9], we gave an explicit method to compute the peripheralstructure π1(BA) → π1(EA), induced by the inclusion map. This is related to the work of E.Hironaka [Hir01] on complexified real arrangements, but the general complex case requires a morecareful study of generators of π1(BA), coming from cycles of the graph. Then we derive from thisperipheral structure a new topological invariant of line arrangements. This invariant is a root ofunity defined for triples composed by an arrangement, a torsion character in C∗ and a cycle in theincidence graph of the arrangement; there are combinatorial restrictions for the availability of thisinvariant, in particular the cycle must be non trivial and satisfy some resonant conditions. It is infact extracted from the homological reduction H1(BA)→ H1(EA), which is a more tractable object,and corresponds to the value of the character on certain homology classes of the boundary manifold,viewed in EA throught the inclusion map. This construction has similarities with knot theory, andthe invariant is a sort of analogue of linking numbers. We obtain an ordered-oriented topologicalinvariant.

As a second main contribution of this work, we give an explicit method to compute the invariant,in terms of braid monodromy. We use braided wiring diagrams introduced by Arvola [Arv92] (seealso Suciu- Cohen [CS97]) to encode the braid monodromy relative to a generic projection ofthe arrangement. Note that this invariant is most probably of algebraic nature, even though ourcomputations are topological.

It appears that this invariant is crucial for the computation of the quasi-projective depth ofa (torsion) character in [AB14]. The knowledge of depths of all characters is equivalent to theknowledge of characteristic varieties; the depth can be decomposed into a projective term anda quasi-projective term, vanishing for characters that ramify along all the lines. An algorithm tocompute the projective part was given by Libgober. An explicit way to compute the quasi-projectivedepth of resonant (torsion) characters is given in [AB14], and it happens that our invariant is crucialfor that method. Hence, our invariant may help to find examples of combinatorially equivalentarrangements with different structure for their characteristic varieties, though we have failed till


now in finding such arrangements. But as we show in this paper, this invariant is interesting in itsown. We compute the invariant for MacLane arrangements with an additional line, and observe thatit takes different values for the two deformation classes. This shows that it provides informationon their topologies, not contained in the combinatorics. In particular, there is no ordered-orientedhomeomorphism between both realizations; note that this fact is a consequence of the same resultfor MacLane arrangements (as shown by Rybnikov).

3.1. Combinatorics and realizations.3.1.1. The incidence graph.

Definition 3.1. A combinatorial type, or simply a (line) combinatorics, is a couple C = (L,P),where L is a finite set and P ⊂ P(L), satisfying that:

• For all p ∈ P, ]p ≥ 2;• For any `1, `2 ∈ L, `1 6= `2, ∃!p ∈ P such that `1, `2 ∈ p.

An ordered combinatorics C is a combinatorics where L is an ordered set.

This notion encodes the intersection pattern of a collection of lines in a projective planes, see Sec-tion 3.1.2, where the relation ∈ corresponds to the dual plane. There are several ways to encode aline combinatorics. For more details, see [ACCAM05].

Definition 3.2. The incidence graph ΓC of a line combinatorics C = (L,P) is a non-orientedbipartite graph where the set of vertices V (C) decomposes as VP (C)q VL(C), with:

VP (C) = vp | p ∈ P, and VL(C) = v` | ` ∈ L.The vertices of VP (C) are called point-vertices and those of VL(C) are called line-vertices. An edgeof ΓC joins v` to vp if and only if ` ∈ p. Such an edge is denoted by e(`, p).

For a line arrangement in the projective plane the incidence graph is the dual graph of thedivisor obtained by the preimage of the line arrangement in the blowing-up of the projective planealong the set of multiples points of the arrangement, see §3.1.2.

3.1.2. Realization.Let A be a line arrangement in CP 2 and let PA be the set of multiple points of A; then

CA := (A,PA) is the combinatorics of A. Given a combinatorics C = (L,P), a complex realizationof C is a line arrangement A in CP 2 such that its combinatorics agrees with C. An ordered realizationof an ordered combinatorics is defined accordingly. The existence of realizations of a combinatoricsdepends on the field Z.

Given a line arrangement A in CP 2, or more generally a set of irreducible curves A in a projectivesurface X, we will denote

⋃A the union of those curves.

Remark 3.3. Let A be a line arrangement in CP 2 with combinatorics C = (L,P) (note that

L = A). For ` ∈ L and p ∈ P the notion ` ∈ p can be understood in the dual plane CP 2. For

convenience we also use the notation p ∈ `. Let π : CP2 → CP 2 be the composition of the blow-ups

of the points in P; then A := π−1(A) defines a normal crossing divisor in CP2

whose dual graph is

exactly ΓC . Note also that MA := CP 2 \⋃A = CP2 \⋃ A. Let us identify each v` ∈ VL(C) with

a meridian of ` as an element in H1(MA;Z). Note that VL(C) generates H1(MA;Z) and the only


relation satisfied by them is (3.2). Moreover, if vp ∈ VP (C) is identified with a meridian of π−1(P )as element of H1(MA;Z), then, the equality

(3.1) vp =∏p∈`

v` =∏`∈p

v`

holds. Via this identification the space of characters of C coincides with

H1(MA;C∗) = Hom(H1(MA;Z),C∗) ∼= (C∗)]A−1.

Equation (3.1) agrees with Definition 3.10.

The space MA is not compact and this may cause some trouble. To avoid this complication,let Tub(A) be a compact regular neighbourhood of A in CP 2, and let EA := CP 2 \ Tub(A) be theexterior of the arrangement. This is an oriented 4-manifold with boundary such that the inclusionEA →MA is a homotopy equivalence and in particular H1(EA;Z) ≡ H1(MA;Z). As we have seen,if ]A = n + 1, then H1(EA;Z) ∼= Zn and it is freely generated by the meridians of any subset of nlines in A.

We use the notation ΓA for the incidence graph of ΓC .

Example 3.4.

L0 = z = 0 , L1 = −(i+ 2)x+ (2i+ 3)y = 0 , L2 = −x+ (i+ 2)y = 0 ,L3 = −x+ 3y + iz = 0 , L4 = −x+ (2i+ 2)y = 0 .

L0

P0,1

P0,2

P0,4

P0,3 L3

L4

L2

L1 P1,2,4

P1,3

P2,3

P3,4

3.1.3. Neighborhoods and boundary manifold.Let A be a line arrangement with combinatorics C. Let us describe how to construct a compact

regular neighbourhood Tub(A). There are several ways to define it, see [CS08, Dur83], and theyproduce isotopic results.

For each ` ∈ A we consider a tubular neighborhood Tub(`) of ` ⊂ CP 2 and for each p ∈ P weconsider a closed 4-ball Bp centered at p.

Definition 3.5. We say that the set Tub(`) | ` ∈ A ∪ Bp | p ∈ P is a compatible system ofneighborhoods in A if ∀`1 6= `2 we have

Tub(`1) ∩ Tub(`2) = Bp, p = `1 ∩ `2


and the balls are pairwise disjoint. The union of these neighborhoods is a regular neighborhood ofA. Given such a system, for each ` we define the holed neighborhood

N (`) := Tub(`) \⋃p∈`

Bp.

For each ` ∈ A we denote by

ˇ := ` \⋃p∈`

Bp.

This is a punctured sphere (which as many punctures as the number of multiple points in `) andthe space N (`) is (non-naturally) homeomorphic to ˇ×D2, where D2 is a closed disk in C.

Let us consider now π : CP2 → CP 2. For each p ∈ P, π−1(Bp) is a tubular neighborhood of the

rational curve Ep := π−1(p); this is a locally trivial D2-bundle, but not trivial. Note that π−1(N (`))

is naturally isomorphic to N (`) and that Tub(A) := π−1(Tub(A)) is a regular neighborhood of Aobtained by plumbing the tubular neighborhood of its irreducible components.

If p ∈ ` then V`,p := N (`) ∩ Bp is a tubular neighborhood of the trivial knot ` ∩ ∂Bp ⊂ ∂Bp, i.e.a solid torus.

Definition 3.6. The boundary manifold of A is the common boundary BA = EA ∩ Tub(A) ofthe manifolds EA and Tub(A).

Note that BA can be identified with the boundary of Tub(A). This manifold is a graph manifoldobtained by gluing the following pieces:

B` := ∂N(`) \⋃p∈`

V`,p, Bp := ∂Bp \⋃`∈p

V`,p.

In particular, the graph structure of BA is modeled on the incidence graph and determined by thecombinatorics. Note also that BA → Tub(A) \⋃A is a homotopy equivalence.

3.1.4. Nearby cycles.Let A be an arrangement, and γ be a circular cycle of ΓA. The support of γ is defined as:

supp(γ) = ` ∈ A | v` ∈ γ = `1, . . . , `r .with cyclic order `1 < · · · < `r and `r+1 := `1. Let pj := `j ∩ `j+1, j = 1, . . . , r.

Definition 3.7. An embedding of γ in A is a simple closed loop r(γ) ⊂ ⋃A defined as follows.Take a point qj ∈ ˇ

j (qr+1 := q1), j = 1, . . . , r. We denote by pαj a point in `j ∩ Bpj and by pωj apoint in `j+1 ∩ Bpj . Let ραj be a radius in `j from pj to pαj and ρωj be a radius in `j+1 from pj to

pωj . Pick up arbitrary simple paths αj from qj to pαj in ˇj , j = 1, . . . , r, and ωj from pωj−1 to qj ,

j = 2, . . . , r + 1. Then:

r(γ) := α1 · (ρα1 )−1 · ρω1 · ω2 · α2 · . . . · (ραr )−1 · ρωr · ωr+1.

Definition 3.8. A nearby cycle γ associated with γ is a smooth path in BA, homologous to anembedding r(γ) of γ in Tub(A), lying in r⋃

j=1

N(`j) ∪r⋃j=1

Bpj

\⋃A.


Note that if p 6= pj a nearby cycle γ can intersect Bp only at some V`k,p for some k; there arealways nearby cycles which do not intersect Bp for p 6= pj .

3.2. A topological invariant.3.2.1. Characters.

Definition 3.9. A character on a line combinatorics (L,P) is a map ξ : L → C∗ such that

(3.2)∏`∈L

ξ(`) = 1.

A torsion character on a line combinatorics (L,P) is a character ξ where for all ` ∈ L, ξ(`) is a rootof unity.

Namely, we are associating a non-zero complex number to each element of L, such that theproduct of all of them equals 1. These characters have a cohomological meaning for line arrangementsin the complex projective plane.

Definition 3.10. Let ξ be a character on a line combinatorics C = (L,P). For each p ∈ P, wedefine ξ(p) :=

∏`∈p ξ(`).

A cycle of ΓC is an element of H1(ΓC).

3.2.2. Inner cyclic triplets.

Definition 3.11. An inner cyclic triplet (C, ξ, γ) is a line combinatorics C = (L,P), a torsioncharacter ξ on L and a cycle γ of ΓC such that:

(1) for all line-vertex v` of γ, ξ(`) = 1,(2) for all point-vertex vp of γ, and for all ` ∈ p, ξ(`) = 1,(3) for all p ∈ `, with v` ∈ γ, ξ(p) = 1.

The above conditions can be understood in a shorter way: all the vertices of γ and all theirneighbors come from elements m ∈ V (C) such that χ(m) = 1.

Definition 3.12. Let (C, ξ, γ) be a triplet, where C is an ordered combinatorics, ξ a characterand γ a circular cycle of ΓC . A realization of (C, ξ, γ) is a triplet (A, ξA, γA):

• An ordered realization A of C;• A character ξA : H1(EA;Z) → C∗ such that ξA(v`) = ξ(`) under the identification of

Remark 3.3.• A cycle γA in ΓA which coincides with γ via the natural identification ΓA ≡ ΓC .

Due to this natural identifications we usually drop the subindex A. If (C, ξ, γ) is an inner cyclictriplet, then the realization (A, ξ, γ) is inner cyclic.

3.2.3. Definition of I. Let (C, ξ, γ) be a inner-cyclic triplet; and suppose that (A, ξ, γ) is arealization. Denote by i the inclusion map of the boundary manifold in the exterior, i.e., i : BA →EA. We consider the following composition map:

χ(A,ξ) : H1(BA)i∗−→ H1(EA)

ξ−→ C∗

Let γ be a nearby cycle associated with γ. We define

I(A, ξ, γ) := χ(A,ξ)(γ).


Theorem 3.13. [10] If (A, ξ, γ) and (A′, ξ, γ) are two inner-cyclic realizations of (C, ξ, γ) withthe same (oriented and ordered) topological type, then

I(A, ξ, γ) = I(A′, ξ, γ).

The proof of Theorem 3.13 has two aspects. We first show that for a given inner-cyclic realization(A, ξ, γ), the image of a nearby cycle γ by χ(A,ξ) depends only in γ. Hence I(A, ξ, γ) is well-defined for a particular A. Next we study the behavior of this invariant under orientation and orderpreserving homeomorphisms. We prove that there are nearby cycles of the first realization whichare sent to nearby cycles of the second realization; note that we do not need the stronger result thatwould say that the image of any nearby cycle is a nearby cycle.

L0 =∞

L1L2L3

L4

L5

L6

Figure 1. The Ceva-7 arrangement C7

Example 3.14. Let C = (L,P) be the combinatorics defined by L = L0, · · · , L6 and

P = L0, L1, L2 , L0, L3 , L0, L4, L5 , L0, L6 , L1, L3, L5 ,L1, L4, L6 , L2, L3, L4 , L2, L5, L6 , L3, L6 .

The following triplet (C, ξ, γ) is inner-cyclic:

• ξ is the character on C = (L,P) defined by:

(L0, L1, · · · , L6) 7−→ (1,−1,−1, 1,−1,−1, 1).

• γ is the cycle of ΓC defined by vL0 → vP0,3 → vL3 → vP3,6 → vL6 → vP0,6 → vL0 .

The arrangement C7 pictured in Figure 1 is –up to projective transformation– the only onerealization of the combinatorics C. There is a nearby cycle γ associated with the cycle γ suchthat its image by the map i∗ : H1(BC7) → H1(EC7) is −v5 (an algorithm to compute is given inSection 3.4). Then we have:

I(C7, ξ, γ), ξ) = ξ(−v5) = −1.


3.3. Characteristic varieties. The characteristic varieties of an arrangement A are definedas Vk(A) = Vk(MA) and the depth of a character ξ ∈ T(A) is:

depth(ξ) = max k ∈ N | ξ ∈ Vk = dimCH1(MA;C(ξ)).

Hence for the study of characteristic varieties, we need to be able to compute the twistedcohomology spaces of MA.

3.3.1. Geometric interpretation of the notion of inner cyclic.

Let us recall that π : CP2 → CP 2 is the blow-up of CP 2 over the points of P; the main goal

of the construction of CP2

is to obtain A := π−1(A) as a normal crossing divisor (in fact, we needonly to blow up the points of multiplicity at least three, but it is harmless to do extra blow-ups).

By construction, we have that CP2 \ ⋃ A ≡ EA, then a character ξ on π1(EA) can be view as a

character on π1(CP2 \⋃ A) (also noted ξ). Let ΓA be the dual graph of A.

Definition 3.15. A component H ∈ A is unramified for the character ξ if ξ(vH) = 1. It is

inner unramified for ξ if it is unramified and all its neighbours (in ΓA) too. The set of all the

inner unramified components of A is denoted by Uξ ⊂ A. The dual graph of A of inner unramified

components is denoted by ΓUξ .

Definition 3.16. Let A be an arrangement. A character ξ is inner-cyclic if it is torsion and

b1(ΓUξ) > 0.

Proposition 3.17. An inner-cyclic arrangement is the data of a triple (A, ξ, γ), where ξ is an

inner-cyclic character on EA and γ a cycle of ΓUξ .

Note that with this point of view, a nearby cycle is a cycle leaving in the boundary of a regularneighbourhood of the union of inner unramified components.

3.3.2. Quasi-projective depth.The depth of a torsion character is decomposed in two terms, the projective and quasi-projective

depth, see [Art14, Gue13]. Summarizing, if ξ is a torsion character of order n, then there is an

n-fold unbranched ramified quasi-projective cover ρ : EξA → EA where σ : EξA → EξA generates thedeck group of the cover. There is a natural isomorphism from H1(EA;C(ξ)) to the eigenspace of

H1(EξA;C) for σ with eigenvalue e2iπn (See Chapter 1 Example 1.2).

Let ρ : Xξ → CP2

be a smooth model of the projectivization of ρ, where σ generates the deck

group. The inclusion jN : EξA → Xξ induces an injection

j∗N : H1(Xξ;C)→ H1(EξA;C).

We denote by j∗N,ξ the restriction of j∗N to the eigenspaces for σ, σ with eigenvalue e2iπn .

Definition 3.18. Let ξ be a torsion character on π1(EA). The projective depth of ξ is dim imj∗N,ξwhile the quasi-projective depth of ξ is:

deph(ξ) = dim coker(j∗N,ξ).

There are known formulas for the computation of the projective depth given by Libgober,see [Art14] for details. Moreover, there is a finite-time algorithm to compute this projective depth


for any character. We now briefly explain the method of [AB14] to compute the quasi-projectivedepth.

We construct a matrix for a twisted hermitian intersection form · in a vector space having asbase the elements of Uξ; for an arbitrary order of this basis we consider a square matrix Aξ ofsize #Uξ, where coefficients are indexed by elements of Uξ. This matrix depends on the choice of

a maximal tree TUξ of ΓUξ (maybe a maximal forest, since ΓUξ is non necessarily connected). For

each (oriented) edge e not in TUξ we consider a cycle γe consisting on e and a linear chain of ΓUξconnecting the final end of e with its starting point. Let us denote:

χ(e) :=

I(A, ξ, γe) if e /∈ TUξ1 otherwise.

The coefficient associated to two components E and F is:E · F if E = F∑

e χ(e) if E 6= F.

where the sum runs along all the oriented edges from E to F . Note that since there is at mostone edge between E and F , then the sum

∑e χ(e) is either void (hence it vanishes) or consists of a

single summand.

Theorem 3.19 ([Art14]). [10] Let A be an arrangement, and let ξ be a torsion character onπ1(EA) then:

depth(ξ) = corank(Aξ).

The description of the inclusion map done in [9] (see also Section 3.4) and the result on theinvariant obtain in Section 3.2 allow to compute explicitly the quasi-projective depth of any torsioncharacter.

Example 3.20. In the case of the arrangement C7 with the character ξ defined in Example 3.14,the matrix Aξ is: −1 1 1

1 −1 χ(C7,ξ)(γ)1 χ(C7,ξ)(γ)−1 −1

.

Example 3.14 implies that depth(ξ) = 2. Remark that this result is in harmony with the oneobtained by D. Cohen and A. Suciu in [CS99b].

3.4. Computation of the invariant. The definition of I(A, ξ, γ) is quite clear, and mostprobably it may have a more algebraic description. Nevertheless, its actual definition is topological,and we need concrete models of the topology of A and more specifically of the embedding BA → EAat a homology level, see [9].

In this section we recall the notion of wiring diagram of A and briefly explain how to use itto compute the invariant I(A, ξ, γ). The wiring diagram is a (3, 1)-dimensional model of the pair(CP 2,A) which contains all the multiple points of A. Given a cycle γ in ΓA, we will use the diagramto construct a specific nearby cycle γ and calculate its value by the caracter ξ. This gives a generalmethod to compute the invariant from the equations of A.

Let us fix an arbitrary line `0 ∈ A which will be considered as the line at infinity.


Consider the affine arrangement A0 := A\`0 of C2 ≡ CP 2 \`0; for a projective line ` ∈ A, ` 6= `0we will denote by L the corresponding affine line `\`0. Let π : C2 → C be a linear projection, genericwith respect to A in the sense that for all i ∈ 1, . . . , n, the restriction of π|Li is a homeomorphism.

We choose the coordinates in C2 such that π is the first projection. Suppose that the multiple pointsof A lie in different fibers of π and that their images xi satisify <(x1) < · · · < <(xk).

Consider a smooth path ν : [0, 1]→ C whose image starts from a regular value x0 ∈ π(Tub(L0))and passes through x1, . . . , xk in order with ν(tj) = xj . The braided wiring diagram WA associatedto ν (see [Arv92]) is defined by:

WA =

(t, y) ∈ [0, 1]× C | (ν(t), y) ∈⋃A0

.

The space WA ⊂ [0, 1]×C is a singular braid with n strings labelled according to the lines, whosesingular points correspond to the multiple points of A. Let us fix a generic projection πR : C → Rsuch that the n strings are generic outside the singular points. For u = 0, . . . , k − 1, the wiringdiagram over (tu, tu+1) is identified with a regular braid βu in the braid group Bn with n strands.

Figure 2. Decomposition of a wiring diagram

Associated to any singular fiber π−1(tu), containing the multiple point P = Lp1 ∩· · ·∩Lps(P ), let

τu be the local positive half-twist between the strings p1, . . . , ps(P ) leaving straight the other strings(note that those strings are consecutive).For all u, v ∈ 0, . . . , k with u 6= v, define

βu,v =

βv−1 · τv−1 · βv−2 · . . . · βu+1 · τu+1 · βu u < v(βv,u)−1 v < u

.

The braid βu,v is obtained from WA by taking the sub-singular braid bounded by the singular pointsPu and Pv and replacing the singular crossings by the corresponding local half-twist τ . It should be


sp4

sp3

sp2

sp1

Lp4

Lp3

Lp2

Lp1

Figure 3. The half-twist associated to a singular point of multiplicity 4.

noted that the braided wiring diagram encodes in fact the braid monodromy of A relative to theprojection π. The operation of replacing singular crossings by half twists corresponds to a particularchoice of a geometric basis of π1(C \ x1, . . . , xk), obtained from perturbations of the path ν.

The cycle γ of ΓA can be described by a cyclically ordered sequence of line-vertices vì0 , · · · , vìr .Through the wiring diagram WA, we can see γ directly in A, in a unique way, by identifying its

vertices to their corresponding wires between two multiple points. To produce an embedding r(γ)as in Definition 3.7, we avoid the singular points in the wires, see [9,Section 4]. It is worth noticingthat the way to push γ off A depends on conventions, in particular around the multiple points of Adifferent from Pjq . Different conventions might give another nearby cycle, but whose the image bythe caracter ξ depends only on γ. In order to compute the image of γ by i∗ induced by the inclusionmap BA → EA at homology level, we use an abelian version of [9,Theorem 4.3].

3.5. Example. The MacLane arrangements are two conjugated arrangements coming fromMacLane’s matroid [Mac36]. It is the arrangement with a minimal number of lines such that thecombinatorics admits a realization over C but not over R (see [BLVS+99b, Example 6.6.2(3)]).These arrangements are constructed as follows. Let us consider the 2-dimensional vector space onthe field F3 of three elements. Such a plane contains 9 points and 12 lines, 4 of them pass throughthe origin. Let L be F2

3 \ (0, 0) and P the set of lines in F23. This provides a line combinatorics

(L,P,b), where for all ` ∈ L, P ∈ P, we have P b ` ⇔ (` ∈ P, in F23). Figure 4 represents the

ordered MacLane’s combinatorics viewed in F23. As an ordered combinatorics, it admits two ordered

complex realizations.

• •

• • •

• • •

6 5

2 1 3

4 7 8

Figure 4. Ordered MacLane combinatorics: lines are points in F23 \ (0, 0) and

multiple points are affine lines F23

We can give equations to the realizations:

L1 : y − ζz = 0, L2 : y − z = 0, L3 : y − ζz = 0, L4 : x− z = 0,

L5 : x− ζy = 0, L6 : x− ζy = 0, L7 : x− ζz = 0, L8 : x− ζz = 0,

4. PERSPECTIVES 57

where ζ is a primitive cubic root of unity (its choice determine the realization). Add to MacLanearrangements a line L0 passing through the intersection points: L1 ∩L2 ∩L3 and L4 ∩L7 ∩L8, i.e.,L0 : z = 0 in the above equations. We obtain two ordered realization denoted byM+ andM− andcalled respectively positive and negative extended MacLane arrangement.

Figure 5. Wiring diagram of extended MacLane arrangement

It is not hard to see that the only inner-cyclic characters are ξ and ξ−1 where ξ is defined by:

(v0, v1, . . . , v8) 7−→ (1, ζ, ζ, ζ, ζ2, 1, 1, ζ2, ζ2),

with corresponding cycle γ : vL0 → vP0,6 → vL6 → vP5,6 → vL5 → vP0,5 → vL0 in ΓM+ . We compute

I(M+, ξ, γ) = ζ2, I(M−, ξ, γ)(γ) = ζ.

As a consequence of this paper there is no homeomorphism φ : (CP 2,M+) → (CP 2,M−)preserving orders and orientations of the lines. Note that it is a consequence that this result is alreadytrue for the MacLane arrangements; the fact that MacLane arrangements satisify this property isdone using the techniques in [ACCAM07, Ryb11]) and the invariant in our work does not providean obstruction.

4. Perspectives

4.1. Braid group representations and polynomial invariants. The starting point of thisproject in collaboration with J.I.Cogolludo is an invariant of equisingular families of curves, con-structed by Libgober [Lib89] by composing the braid monodromy to a braid group representation.In the case of the Burau representation, the invariant coincides with the Alexander polynomial ofthe curve. Historic and examples are given in [5]. The proof uses Fox calculus, but we can also proveit with Milnor-Reidemeister torsion with a ”cut and paste” description of the curve complement.


This showes indirectly that the invariant depends only on the fundamental group, in this particularcase.

It is worth noticing that the braid monodromy is induced by the choice of a generic projection,see Section 1. However, it might be intersecting to consider non-generic projections. For example,an irreducible component of degree 1 of the curve could appear as a fiber of the projection.

Question 4.1. For generic projections and associated braid monodromies, does this invariantdepend only on the fundamental group? What about non-generic projections?

Notice that in contrary to knot invariants (and Markov theorem), there are no constraint on thecompatibility between the representation on different numbers of strings.

We want to consider a familly of linear representations of the braid groups that factor throughtthe Hecke algebras [Jon87]. Let q be a complex number. Let L be a link in S3, obtained as theclosure of a braid σ ∈ Bn. The original definition of the Jones polynomial is a sum

VL(q) =(−1)n(

√q)e−n+1

1− q2

[n2

]∑m=0

(1− qn−2m+1)tr(πYm(σ))

indexed 2-rows Young diagrams, where e depends only on σ. This sum is a specialisation of thedecomposition of the Ocneanu trace (giving the HOMFLY polynomial) as a sum of characterscorresponding to irreducible representations of the Hecke algebras (if q = 1, they are simply thealgebras over the symmetric groups).

Our plan is to apply this sum to define an invariant J of equisingular families of curves. We firstwant to compute it in some examples. Notice that if the number of strings appearing in the braidmonodromy is big, the explicit matrices are difficult to compute. Up to 4 strings, the representationsare determined by Burau and (a specialisation of) the Lawrence-Krammer representation, whosematrices are well-known. For curves of small degree, considering non-generic projections would allowto work with a very small number of strings.

Question 4.2. Is J related to the Jones polynomials of the local links of the singular points ofthe curve?

Going back to Alexander polynomial, another specialisation of the HOMFLY polynomial is asum indexed by the Young diagrams of the form k + 1 + 1 + · · ·+ 1 = n. It can be shown that therepresentations of Bn with these diagrams are the exterior powers of the Burau representation ρ -upto a sign-. We get, up to a unit,

∆L(q) =1− t1− tn

n−1∑k=0

tr(πYk(σ)) =1− t1− tn

n−1∑k=0

(−1)ktr(Λkρ(σ)

)=

1− t1− tn det(1− ρ(σ)),

where the last equality follows from the Mac Mahon theorem that expresses in general the carac-teristic polynomial of a matrix as an expansion of the traces of its exterior powers.

This is related to the projects of Chapter 4, Section 6.2.

4.2. Alexander polynomials of line arrangements. A coloring 1 of a link diagram is anassignment of colors to the arcs of the diagram. The existence or not of a 3-coloring with certain

1In this section, the notion of ”coloring” is different than in the rest of the memoire, where ”colored links” areconsidered. We decided however to keep this terminology for historical reasons.

4. PERSPECTIVES 59

constraints at the crossings is a knot invariant introduced by Fox [Fox62a, Fox70]. This can begeneralized to any number of colors, or even to continuous palettes [SW00]. These invariants arein fact related to the homology of branched cyclic coverings, or even to the Alexander polynomial,derived from the fundamental group. This can be shown with the Wirtinger presentation. Thedefinition of the Alexander polynomial by Alexander himself were given with other kind of coloring,by considering crossings and regions. The equivalence can be find in [CSW14].

In the case of line arrangements, a sort of analogue of knot diagrams is given by wiring diagrams,see Section 3.4. Recall that they were introduced by Arvola [Arv92], and enriched by colorings tocompute a presentation of the fundamental group (similarly to Wirtinger method).

Let A be an arrangement in CP 2 with complement MA = CP 2 \A, and ε : H1(MA)→ Z be anepimorphism. Consider the Alexander polynomial ∆(A, ε) = ∆0(Hε

1(MA,Z[t±1])).

Problem 4.3. Find a combinatorial description of ∆(A, ε) from the wiring diagram.

The expected model will be very similar to the one developped in Chapter 4 Section 3.0.3. Wedefine a planar algebra ArrZ as in Chapter 4 Section 5.3. To define cobordisms in our context, weconsider the following construction. Let A be an arrangement in CP 2. Let B1, . . . , Bp be disjoints4-balls in the interior of a ball B in CP 2. For every i ∈ 0, . . . , p, let Li be the oriented link A∩∂Bi.The intersection A with B \ B1, . . . , Bp is called a cobordism. We obtain an operad very similarto the operad of Definition 5.6 but links are not always trivial and cobordisms are peaces of linearrangements. In this setting, a tangle is any pair (B4, F ) where F is the intersection of a 4-ballwith an arrangement.

Question 4.4. Caracterise the links arising as the intersection of a 3-sphere with a line ar-rangement in CP 2. Can one caracterise the Alexander modules of such links?

Then, the plan is to construct ArrZ as a morphism of the operad above and the operad Hom ofZ[t±1]-modules. Once this algebra is constructed, any wiring diagram of an arrangement will allowto decribe the arrangement as a collection of tangles and cobordisms.

CHAPTER 4

Functorial extensions of the abelian Reidemeister torsion

The Topological Quantum Field Theories are a very fruitful line of investigation relating geometryand physics, coming originally from ideas of quantum physics. In the eighties, Witten clarified theirrole in the study of topological invariants of knots and manifolds (in particular, the Jones polynomialof knots). Then Atiyah [Ati88] presented a set of axioms for TQFT which gave a mathematicalformalization of Witten’s constructions on invariants of four-dimensional manifolds coming from aquantum field theory (known as Donaldson Theory). Nowadays, TQFT are most commonly viewedas a functor from a certain category of cobordisms to a category of modules or vector spaces. Thesearch for functors of this kind that enclose known topological invariants of manifolds has been untilnow an extremely active area of mathematics.

The problem of extending the Alexander polynomial to a functor from a category of cobor-disms has been solved by Frohman and Nicas. They used elementary intersection theory in U(1)-representation varieties of surfaces [FN91]. (See also [FN94] for a much more general constructionusing PU(N)-representations.) Later, Kerler showed that the Frohman-Nicas functor is in factequivalent to a TQFT based on a certain quasitriangular Hopf algebra [Ker03]. The Alexanderpolynomial of a knot K in an integral homology 3-sphere N is recovered from this functor by takingthe ‘graded’ trace of the endomorphism associated to the cobordism that one obtains by “cutting”N \K along a Seifert surface of K. Note that the way how their functor determines the Alexanderpolynomial is somehow extrinsic, in that it goes through the choice of a Seifert surface.

This chapter is devoted to a new functorial model for the Alexander polynomial, and moregenerally for the abelian Reidemeister torsion (the Milnor-Turaev torsion, see Chapter 1) in differentcontexts. We describe a topological construction that starts from the Alexander function introducedby C.Lescop [Les98] and provides a functor for 3-dimensional cobordisms and tangles, among others.Our functor intrinsically contains the Alexander polynomial of oriented knots in oriented integralhomology 3-spheres, by considering any knot of this type as a ’bottom knot’ in the style of [Hab06].It specializes, for some specific twisted coefficients of the homology, to the Frohman–Nicas functor.We also observe that our construction might be related to the TQFT introduced by Blanchet,Costantino, Geer and Patureau-Mirand [BCGPM16].

The Magnus representations of mapping class groups appear naturally in our constructions. IfΣ is a compact connected oriented surface with ∂Σ 6= ∅, the mapping class group MCG(Σ) consistsof the isotopy classes of (orientation-preserving) self-homeomorphisms of Σ fixing the boundarypointwise. Magnus representations usually refer to those representations of subgroups of MCG(Σ)that are defined by assigning to an f ∈ MCG(Σ) the matrix with entries in Z[π] consisting of Fox’sfree derivatives of f∗ : π → π with respect to a fixed basis of π. Thus they have a group-theoreticaldefinition, which goes through the automorphism group of π. Birman [Bir74] gave an algebraicexposition and survey of these representations, explaining for instance how far they are from being

61

62 4. FUNCTORIAL EXTENSIONS OF THE ABELIAN REIDEMEISTER TORSION

group homomorphisms, or analyzing their kernels and images. One of her motivations was to givea unified treatment of the Burau representation of the braid group and the Gassner representationof the pure braid group. These correspond to the case where the surface Σ is a disk with markedpoints or holes, and are defined from the Magnus representations by reducing the coefficients in Z[π]to some appropriate commutative rings.

The Gassner representation of the pure braid group was later extended to string links (also called“pure tangles”) by Le Dimet [LD92]. Kirk, Livingston and Wang gave a topological interpretationof this extension and a simple proof of its invariance under concordance [KW01]. Their approachis based on a natural action of the monoid of string links on the twisted homology of a punctureddisk, and relies on the topological interpretation of Fox’s free derivatives in terms of universal covers.More recently, Cimasoni and Turaev extended the Burau and Gassner representations to arbitrarytangles [CT05, CT06]. Their invariant is defined as a functor from the category of (colored) tanglesto the category of “Lagrangian relations” between skew-Hermitian modules.

In the case of a surface Σ of positive genus, Magnus representations have been used and studiedby Morita [Mor93], Suzuki [Suz03] and Perron [Per06] among others. In this case too, there is agroup-theoretical definition in terms of Fox’s free derivatives as well as a topological definition usingtwisted homology (see [Suz05]). Furthermore, the Magnus representation is extended in [Sak08]to the monoid of homology cobordisms, which is a natural higher-genus version of the monoid ofstring links. We refer to Sakasai [Sak12a] for an overview of these topics.

In the first section, we introduce the categories CobG of 3-dimensional cobordisms enriched witha representation of their fundamental group to an abelian group G, and Tangµ for µ-colored tangles.Then, we recall the construction by V.Jones of a planar operad of tangles. In the second section,we construct the monoidal Reidemeister functor R from CobG to the cateogry of F-vector spaces,where F is a field and G a subgroup of its units. We show how R restricted to homology cobordismscarries more information than the Magnus representation. Then we show that R specializes to anAlexander functor A on CobG, that can also be directly constructed from the Alexander function.Finally, we briefly explain how the same objects can be applied to construct an Alexander functorB on the category Tangµ. The third section is devoted to Alexander polynomials. We explain howA contains the Alexander polynomial of a knot in a 3-homology sphere and of a closed 3-manifold,and B contains the Alexander polynomial of the closure of a (1, 1)-tangle. Then we proceed to adiagrammatic (local) model for the Alexander polynomial of a link in S3. By using similar toolsto our previous constructions, we define a planar algebra that provides a topological point of viewon planar algebras considered by D.Bar-Natan and J.Archibald [Arc]. The forth section is devotedto the construction of a functor Mag. Inspired by the work of Cimasoni and Turaev on tangles,Mag takes values in a category of Lagrangian subspaces of Hermitian modules. It appears that Magis contained in A (the details on this result are stated in [12]). A main difficulty of this work isthe existence of a hermitian twisted intersection form on surfaces, whose construction is given indetails. The last section should be viewed as a first step of a larger program whose main motivationis to extend these functorial construction to 4-dimensional cobordisms and manifolds. We restrictto surfaces in B4 with ribbon singularities. The next step will be devoted to algebraic curves inCP 2, as it is explained in Chapter 3 Section 4.2 for line arrangements.

1. MANIFOLDS AND CATEGORIES 63

1. Manifolds and categories

1.1. Three-manifolds and cobordisms. We first recall the definition of the category Cob of3-dimensional cobordisms introduced by Crane and Yetter [CY99]. The objects of Cob are non-negative integers g ≥ 0: the object g refers to a compact, connected, oriented surface Fg of genusg with one boundary component. The surface Fg is fixed and will play the role of “model” surface.Furthermore, we assume that the boundary component ∂Fg is identified with S1 and a base point? ∈ S1 = ∂Fg is fixed.

For any integers g+, g− ≥ 0, a morphism g− → g+ in Cob is a cobordism from the surface Fg−to the surface Fg+ : specifically, this is an equivalence class of pairs (M,m) consisting of a compact,connected, oriented 3-manifold M and an orientation-preserving homeomorphism m : F (g−, g+)→∂M , where

F (g−, g+) := −Fg− ∪S1×−1(S1 × [−1, 1]

)∪S1×1 Fg+ ;

here two cobordisms (M,m) and (M ′,m′) are said to be equivalent if there exists a homeomorphismf : M → M ′ such that m′ = f |∂M m. Let m± : Fg± → M be the composition of m|Fg± with the

inclusion of ∂M into M , and set ∂±M := m±(Fg±).

∂+M

∂−M

M

m+

m−

Fg+

Fg−

In the sequel, we will denote a cobordism simply by an upper-case letter M,N, . . . meaningthat the boundary-parametrization is denoted by the corresponding lower-case letter m,n, . . . Thecomposition N M of two cobordisms M ∈ Cob(g−, g+) and N ∈ Cob(h−, h+) is defined wheng+ = h− by gluing N “on the top of” M , i.e. ∂+M is identified to ∂−N using the boundaryparametrizations m+ and n−:

N

n+

n−

Fh+

Fh−

M

m+

m−

Fg+

Fg−

:=

Fh+

Fg−

N

M

m−

n+

For any integer k ≥ 0, the identity of the object k in Cob is the cylinder Fk × [−1, 1] with theboundary-parametrization defined by the identity maps.

The category Cob can be enriched with a strict monoidal structure [CY99]. We assume that, forany integer g ≥ 1, the model surface Fg is constructed by doing the iterated boundary-connected sumof g copies of the model surface F1 in genus 1. Thus, for any g, k ≥ 0, the boundary-connected sum


Fg ]∂ Fk is identified with Fg+k. The tensor product in the category Cob is defined by g⊗ k := g+ kat the level of objects, and it is defined by M ⊗N := M]∂N at the level of morphisms:

M

m+

m−

Fg+

Fg−

⊗ N

n+

n−

Fh+

Fh−

:= M N

m+]∂n+

m−]∂n−Fg−+h−

Fg++h+

Let G be an abelian group. We now define the category CobG of 3-dimensional cobordisms withG-representations following [8]. The objects of CobG are pairs (g, ϕ) consisting of an integer g ≥ 0and a group homomorphism ϕ : H1(Fg) → G. A morphism (g−, ϕ−) → (g+, ϕ+) in CobG is a pair(M,ϕ) consisting of a cobordism M ∈ Cob(g−, g+) and a group homomorphism ϕ : H1(M) → Gsuch that ϕ m± = ϕ±. The composition of two morphisms (M,ϕ) ∈ CobG((g−, ϕ−), (g+, ϕ+)) and(N,ψ) ∈ CobG((h−, ψ−), (h+, ψ+)) such that (g+, ϕ+) = (h−, ψ−), is defined by

(N,ψ) (M,ϕ) := (N M,ψ + ϕ)

where N M is the composition in Cob and ψ + ϕ : H1(N M) → G is defined from ϕ and ψ byusing the Mayer–Vietoris theorem.

The strict monoidal structure of Cob extends to CobG in the following way. The tensor productof objects is defined by (g, ϕ)⊗ (h, ψ) := (g + h, ϕ⊕ ψ) where H1(Fg+h) = H1(Fg]∂Fh) is identifiedwith H1(Fg)⊕H1(Fh) using the Mayer–Vietoris theorem; the tensor product of morphisms is definedby (M,ϕ) ⊗ (N,ψ) := (M]∂N,ϕ ⊕ ψ) where H1(M]∂N) is identified with H1(M) ⊕ H1(N) usingthe Mayer–Vietoris theorem again.

1.2. Links and tangles. Tangles may be thought of as small pieces or local pictures of knotsand links. They can be glued together in different ways, inducing different algebraic structures. Inthis section, we recall two classical point of views. The first consider tangles as morphisms in acategory Tang, similarly to cobordisms in Section 1.1.

Then, we present tangles in the more recent point of view of planar algebras, introduced byV.Jones [Jon99].

1.2.1. The category Tangµ. Let D2 be the closed unit disk in C2. Given a positive integer n,

denote by x1 < · · · < xn a collection of points ofD2, in the real line. Let ε and ε′ be sequences of±1 ofrespective length n and n′. An (ε−, ε+)-tangle is the pair consisting of the cylinder D2× [0, 1] and its

oriented smooth 1-submanifold τ whose oriented boundary ∂τ is∑n′

j=1(ε+)j(x′j , 1)−∑n

i=1(ε−)i(xi, 0).

Note that we must have∑

i(ε−)i =∑

j(ε+)j .

Two (ε−, ε+)-tangles (D2 × [0, 1], τ1) and (D2 × [0, 1], τ2) are isotopic if there exists an auto-homeomorphism h of D2 × [0, 1], keeping D2 × 0, 1 fixed, such that h(τ1) = τ2 and h|τ1 : τ1 ' τ2

is orientation-preserving. We now consider tangles up to isotopy. Given a (ε−, ε+)-tangle τ1 anda (ν−, ν+)-tangle τ2 with ε+ = ν−, their composition is the (ε−, ν+)-tangle τ2 τ1 obtained bygluing the two cylinders along the disk corresponding to ε+ and shrinking the length of the resultingcylinder by a factor 2 (see Figure 1).

1. MANIFOLDS AND CATEGORIES 65

Figure 1. Composition of tangles in Tang

The category of oriented tangles Tang is defined as follows: an object is a finite sequence ε of±1, and a morphism ε− → ε+ is the isotopy class of a (ε−, ε+)-tangle.

Let µ be a positive integer and H be the free abelian group generated by t1, . . . , tµ. A µ-coloredtangle is a pair (τ, ψ) where τ is a tangle with exterior Xτ and ψ : H1(Xτ ) → H is a morphism.The category Tangµ of µ-colored tangles is defined as follows. The objects of Tangµ are pairs (ε, ϕ)consisting of a sequence ε and a morphism ϕ : H1(Dn) → H. A morphism (ε−, ϕ−) → (ε+, ϕ+) inTangµ is a µ-colored tangle (τ, ψ) such that the colorings ψ,ϕ± are consistent. The category Tangµhas a monoidal structure similar to the category CobG, see the previous section.

Note that tangles could also be treated with a diagrammatic point of view, via planar projections.

1.2.2. A Planar operad. In this section, tangles are 2-dimensional, and they live in disks insteadof rectangles. Let D be the unit disk in C, and for any positive integer n, let x1, . . . , x2n be a fixedordered set of points in ∂D.

Definition 1.1. Let n be a positive integer. A tangle 1 on n-strands is a proper embeddingτ of an oriented 1-manifold in D. It consists of some copies of the circle and n copies of the unitinterval whose boundary are x1, . . . , x2n. The singular set of τ is a finite number of crossings.

Let µ be a positive integer. A µ-colored tangle is a pair (τ, ψ) where τ is a tangle and ψ is amap from the set of strands and circles to the set t1, . . . , tµ. Two colored tangles are equivalentif they are related by Reidemeister moves respecting the coloring.

Definition 1.2. A p-diagram consists of the following data:

• the unit disk D = D0 in C together with a finite set of disjoint subdisks D1, . . . , Dp in theinterior of D. For every i in 0, . . . , p, each Di have 2ni distinct marked points with signon its boundary (with n = n0), and a base point ∗ on the boundary of each disk.

1We use the same word tangle for two slightly different notions in Section 1.2.1 and Section 1.2.2. The one weuse should be clear from the context.


Figure 2. A tangle on 3-strands.

• a finite set of embedded oriented arcs whose boundary are marked points in the Di. Eachmarked point is the boundary point of some string -which meets the corresponding disktransversally- and the sign is coherent with the orientation.

Figure 3. A 3-diagram.

Definition 1.3. Consider a p′-diagram P ′ and a p′′-diagram P ′′ such that D′i is a disk of P ′

with n′i = n′′, for some i ∈ 1, . . . , p′. If the signs of the marked points match, we define thediagram P = P ′ iP ′′ by rescaling via isotopy the tangle P ′′ so that the boundary of D′′ is identifiedwith the boundary of D′i, and making its marked and base points coincide to those of D′i. Then D′iis removed to obtain P ′ i P ′′. This operation is well defined since the starting points eliminate anyrotational ambiguity.

Two µ-colored diagrams can be composed once the coloring match on the boundary components.The µ-colored diagrams form and operad denoted Dµ.

Given a p-diagram and a collection of tangles τ1, . . . , τp, one may create a new tangle if n(τi) = nifor all i = 1, . . . , p by gluing each τi into the internal disk Di of the diagram. The action of coloreddiagrams on colored tangles is defined once the coloring coincide on the boundary components.

2. The Reidemeister functor

In this Section, we first construct the Reidemeister functor. We use elementary theory of Rei-demeister torsions, but cell chain complexes are not necessarily acyclic. Next, we construct the

2. THE REIDEMEISTER FUNCTOR 67

Alexander functor, based on the ’Alexander function’ introduced by Lescop [Les98]. This functionis defined on an exterior power of a certain Alexander module and it takes values in a ring of Laurentpolynomials. Lescop’s definition proceeds in a rather elementary way using a presentation of theAlexander module.

Since the works of Milnor [Mil66] and Turaev [Tur86], it is known that the Alexander polyno-mial of knots and 3-manifolds can be interpreted as a special kind of abelian Reidemeister torsion.We follows this direction and show that our Reidemeister functor specializes to the Alexander func-tor, under some hypothesis on the coefficients.

Our construction is written in the context of 3-dimensional cobordisms, but works as well forbraids and tangles. In the last Section, we briefly sketch how the Alexander functor can be appliedto this situation.

2.1. Definition of R. In this section, we construct the Reidemeister functor R. We fix afield F and a subgroup G of F×. The extension of a group homomorphism ϕ : A → G to a ringhomomorphism Z[A]→ F is still denoted by ϕ.

Let M be a compact connected orientable 3-manifold with connected boundary, and let ϕ :H1(M) → G be a group homomorphism. We fix a base point ? ∈ ∂M and we set g := g(M) =1− χ(M).

Lemma 2.1. We have Hϕi (M,?) = 0 if i = 0 or i > 2. Moreover, we have

dimHϕ1 (M,?) = g + dimHϕ

2 (M,?).

Denote H := Hϕ1 (M,?) and assume in this paragraph that dimH = g. We choose a cell

decomposition of M where ? is a 0-cell: by Lemma 2.1, the homology of the ϕ-twisted cell chaincomplex Cϕ(M,?) is concentrated in degree 1. For every dimension i ∈ 0, . . . , 3, let ni ≥ 0 be

the number of relative i-cells of (M,?) and order them σ(i)1 , . . . , σ

(i)ni in an arbitrary way. For every

cell σ of (M,?), we also choose an orientation of σ and a lift σ of σ to the maximal abelian cover

M of M . Thus, we get a basis c := (c3, c2, c1, c0) of the F-chain complex Cϕ(M,?) where, for every

i ∈ 0, . . . , 3, the basis of the F-vector space Cϕi (M,?) is given by ci :=(1 ⊗ σ(i)

1 , . . . , 1 ⊗ σ(i)ni

).

Then we consider the function Hg → F defined by

(2.1) (h1, . . . , hg) 7−→τ(Cϕ(M,?); c, (h1, . . . , hg)

)if h1 ∧ · · · ∧ hg 6= 0,

0 otherwise.

Here τ (C; c, h) denotes the torsion of the finite F-chain complex C with basis c and homologicalbasis h. It follows from the definition of the torsion that the map (2.1) is multilinear and alternate.

Definition 2.2. The Reidemeister function of M with coefficients ϕ is the F-linear map RϕM :ΛgH → F defined by (2.1) if dimH = g and by RϕM := 0 if dimH 6= g.

The map RϕM is only defined up to multiplication by an element of ±G ⊂ F. It provides atopological invariant of M .

We construct a functor R from the Reidemeister function R. For this we associate to any object(g, ϕ) of CobG the exterior algebra

R(g, ϕ) := ΛHϕ1 (Fg, ?)


of the F-vector space Hϕ(Fg, ?) = Hϕ1 (Fg, ?), which has dimension 2g. Next, we associate to any

morphism (M,ϕ) from (g−, ϕ−) to (g+, ϕ+) an F-linear map

R(M,ϕ) : ΛHϕ−1 (Fg− , ?) −→ ΛH

ϕ+

1 (Fg+ , ?)

of degree δg := g+ − g− in the following way. We set H := Hϕ1 (M, I) where I := m(? × [−1, 1]),

H± := Hϕ±1 (Fg± , ?) and g := g+ +g−. Then, for any integer j ≥ 0, the image R(M,ϕ)(x) ∈ Λj+δgH+

of any x ∈ ΛjH− is defined by the following property:

∀y ∈ Λg−jH+, RϕM(Λjm−(x) ∧ Λg−jm+(y)

)= ω

(R(M,ϕ)(x) ∧ y

).

Here ω : Λ2g+H+ → F is an arbitrary volume form which is integral in the following sense: regardingH+ as F⊗Z[H1(Fg+ )]H1(Fg+ , ?;Z[H1(Fg+)]), we assume that ω arises from an arbitrary volume form

on the free Z[H1(Fg+)]-module H1(Fg+ , ?;Z[H1(Fg+)]). Due to the choices of this volume form and

of the ordered/oriented lifts of the cells to M , the map R(M,ϕ) is only defined up to multiplicationby an element of ±G ⊂ F. Besides, R(M,ϕ) is trivial on ΛjH− for any j < max(0,−δg) and anyj > min(g, 2g−).

The next two results show that the above paragraph defines a monoidal functor R : CobG →grVectF,±G.

Theorem 2.3. [8] Let (M,ϕ) ∈ CobG((g−, ϕ−), (g+, ϕ+)) and (N,ψ) ∈ CobG((h−, ψ−), (h+, ψ+))be morphisms such that (g+, ϕ+) = (h−, ψ−). We have

R((N,ψ) (M,ϕ)

)= R(N,ψ) R(M,ϕ).

Proposition 2.4. Let (M,ϕ) ∈ CobG((g−, ϕ−), (g+, ϕ+)) and (N,ψ) ∈ CobG((h−, ψ−), (h+, ψ+))be morphisms. We have

R((M,ϕ)⊗ (N,ψ)

)= R(M,ϕ)⊗ R(N,ψ).

2.2. The Magnus representation. Let us now fix an integer k ≥ 1, an abelian group G and agroup homomorphism ψ : H1(Fk)→ G. We shall compute the functor R on the monoid of homologycobordisms over the surface Fk. A homology cobordism over Fk is a morphism M : k → k in thecategory Cob such that m± : H1(Fk) → H1(M) is an isomorphism. The set of equivalence classesof homology cobordisms defines a submonoid

C(Fk) ⊂ Cob(k, k).

We restrict ourselves to homology cobordisms M such that the composition

H1(Fk)m−

'// H1(M)

m−1+

'// H1(Fk)

ψ// G

coincides with ψ. Thus we obtain a submonoid

Cψ(Fk) ⊂ C(Fk),which we also view as a submonoid of CobG

((k, ψ), (k, ψ)

)by equipping every cobordism M of the

above form with the homomorphism ψ := ψ m−1− = ψ m−1

+ : H1(M)→ G.Assume now that G is a multiplicative subgroup of a field F. The extension of ψ : H1(Fk)→ G

to a ring homomorphism Z[H1(Fk)]→ F is still denoted by ψ. We set

Hψ := Hψ1 (Fk, ?)

2. THE REIDEMEISTER FUNCTOR 69

and, when we are given an M ∈ Cψ(Fk), we denote H := Hψ1 (M, I). The fact that the map m± :

H1(Fk)→ H1(M) is an isomorphism of abelian groups implies that m± : Hψ → H is an isomorphismof F-vector spaces. (See [KW01, Proposition 2.1] for a similar statement.) Consequently, we areallowed to set rψ(M) := m−1

+ m− : Hψ → Hψ. This results in a monoid homomorphism

rψ : Cψ(Fk) −→ Aut(Hψ),

which is called the Magnus representation. See [Sak12b] for a survey of this invariant.The Reidemeister functor restricts to a monoid homomorphism

R : Cψ(Fk) −→ grVectF,±G(ΛHψ,ΛHψ

).

We now compute this projective representation of the monoid Cψ(Fk).

Proposition 2.5. For any M ∈ Cψ(Fk) with top surface ∂+M , we have

R(M,ψ) = τψ(M,∂+M) · Λ(rψ(M)

): ΛHψ −→ ΛHψ

where τψ(M,∂+M) is the Reidemeister torsion of (M,∂+M).

2.3. The Alexander functor. In this section, we fix a finitely generated free abelian groupG; the extension of a group homomorphism ϕ : A → G to a ring homomorphism Z[A] → Z[G] isstill denoted by ϕ. Let H be a Z[G]-module of finite type.

Definition 2.6 (Lescop [Les98]). Consider a presentation of H with deficiency g:

(2.2) H = 〈γ1, . . . , γg+r | ρ1, . . . , ρr〉.Let Γ be the Z[G]-module freely generated by the symbols γ1, . . . , γg+r, and regard ρ1, . . . , ρr aselements of Γ. Then the Alexander function of M with coefficients ϕ is the Z[G]-linear map AϕM :ΛgH → Z[G] defined by

AϕM (u1 ∧ · · · ∧ ug) · γ1 ∧ · · · ∧ γg+r = ρ1 ∧ · · · ∧ ρr ∧ u1 ∧ · · · ∧ ug ∈ Λg+rΓ

for any u1, . . . , ug ∈ H, which we lift to some u1, . . . , ug ∈ Γ in an arbitrary way.

The map AϕM can be concretely computed as follows: if one considers the r × (g + r) matrixdefined by the presentation (2.2) of H, and if one adjoins to this matrix some row vectors givingu1, . . . , ug in the generators γ1, . . . , γg+r, then AϕM (u1 ∧ · · · ∧ ug) is the determinant of the resulting(g + r) × (g + r) matrix. It is shown in [Les98, §3.1] that, up to multiplication by a unit of Z[G](i.e., an element of ±G), the map AϕM does not depend on the choice of the presentation (2.2).

Example 2.7. Suppose that G has rank 2 and is generated by t1, t2. Consider the module Hwhose presentation has generators γ1, . . . , γ4 and two relations given by the matrix:(

−1 0 1 00 −1 1− t1 t2

)The values of the Alexander function A : Λ2H → R are

A(γ1 ∧ γ2) = t2, A(γ1 ∧ γ3) = 0, A(γ1 ∧ γ4) = 1,

A(γ2 ∧ γ3) = −t2, A(γ2 ∧ γ4) = t1 − 1, A(γ3 ∧ γ4) = 1.

Let Q(G) be the field of fractions of Z[G]. The following lemma, which is implicit in [Les98],shows that either the Alexander function is trivial or it induces by extension of scalars a volumeform on HQ := Q(G)⊗Z[G] H.


Lemma 2.8. We have dimHQ ≥ g, and AϕM 6= 0 if and only if dimHQ = g.

Let M be a compact connected orientable 3-manifold with connected boundary. We fix a basepoint ? ∈ ∂M and a group homomorphism ϕ : H1(M) → G. The genus of M is the integerg(M) := 1− χ(M), i.e. the genus of the surface ∂M .

Lemma 2.9. There exists a presentation of the Z[G]-module Hϕ1 (M,?) whose deficiency is g(M).

We now simplify our notation by setting g := g(M) and H := Hϕ1 (M,?).

In order to define a functor A, we associate to any object (g, ϕ) of CobG the exterior algebra

A(g, ϕ) := ΛHϕ1 (Fg, ?)

of the Z[G]-module Hϕ(Fg, ?) = Hϕ1 (Fg, ?), which is free of rank 2g. Next, we associate to any

morphism (M,ϕ) ∈ CobG((g−, ϕ−), (g+, ϕ+)

)a Z[G]-linear map

A(M,ϕ) : ΛHϕ−1 (Fg− , ?) −→ ΛH

ϕ+

1 (Fg+ , ?)

of degree δg := g+ − g− as follows. We denote by I the interval m(?× [−1, 1]), which connects thebase point of the bottom surface ∂−M to that of the top surface ∂+M . We set H := Hϕ

1 (M, I),H± := H

ϕ±1 (Fg± , ?) and g := g+ +g−. Then, for any integer j ≥ 0, the image A(M,ϕ)(x) ∈ Λj+δgH+

of any x ∈ ΛjH− is defined by the following property:

∀y ∈ Λg−jH+, AϕM(Λjm−(x) ∧ Λg−jm+(y)

)= ω

(A(M,ϕ)(x) ∧ y

).

Here ω : Λ2g+H+ → Z[G] is an arbitrary volume form on H+. Due to the choices of ω and ofthe presentation of H, the map A(M,ϕ) is only defined up to multiplication by an element of ±G.Besides, observe that A(M,ϕ) is trivial on ΛjH− for any j < max(0,−δg) and any j > min(g, 2g−).

We can show directly that the above paragraph defines a monoidal functor A from CobG togrModZ[G],±G. A proof using Alexander functions is given in [8]. Otherwise, we can prove that A is

a specialisation of R. Suppose that G is a finitely generated free abelian group. Let Q(G) be thefraction field of Z[G].

Theorem 2.10. [8] The following diagram is commutative:

grModZ[G],±G

Q(G)⊗Z[G](−)

||

CobG

A 44

R **

grVectQ(G),±G

Consider now the monoid of homology cobordisms over the surface Fk and the Magnus represen-tation defined in Section 2.2. We denote by ψZ : Z[H1(Fk)]→ Z[G] the extension of ψ : H1(Fk)→ G

to a ring homomorphism and we set HψZ := HψZ

1 (Fk, ?). The Alexander functor restricts to a monoidhomomorphism

A : Cψ(Fk) −→ grModZ[G],±G(ΛHψ

Z ,ΛHψZ).

This projective representation of the monoid Cψ(Fk) is computed as follows.


Proposition 2.11. For any M ∈ Cψ(Fk), we have the commutative diagram

ΛHψZ

A(M,ψ)//

_

ΛHψZ _

ΛHψ

∆ψ(M,∂+M)·Λrψ(M)

// ΛHψ

where ∆ψ(M,∂+M) is the Alexander polynomial of the pair (M,∂+M).

2.4. The case of Tangles. We define an Alexander functor denoted B from Tangµ to the

category of S-modules, similarly to the previous Section, where S = Z[H] = Z[t±11 , . . . , t±1

µ ]. DenoteH the free abelian group generated by t1, . . . , tµ.

Let (ε, ϕ) be an object of Tangµ. Let e1, . . . , en be simple oriented loops in Dn turning oncearound xi counterclockwise if εi = +1 and clockwise if εi = −1. Consider the morphism ϕε :π1(Dn)→ H, ei 7→ ϕ(ei). We associate to the object (ε, ϕ) the exterior algebra

B(ε, ϕ) := ΛHϕε1 (Dn;S)

Let (τ, ψ) : (ε−, ϕ−) → (ε+, ϕ+) be a morphism in Tangµ. Set H = Hψ1 (Xτ ;S). We need the

following key lemma:

Lemma 2.12. If (τ, ψ) is a morphism (ε−, ϕ−)→ (ε+, ϕ+) where ε± have length n±, then there

exists a presentation of H with deficiency n = n−+n+

2 .

Using the Alexander function ΛnH → S derived from such a presentation, we can define thefunctor on the morphism τ . We get a linear map of degree (n+ − n−)/2:

B(τ, ψ) : ΛHϕε−1 (Dn− ;S) −→ ΛH

ϕε+1 (Dn+ ;S).

This functor coincides on braids with exterior powers of the reduced colored Burau representation,and on string links with its generalisation by LeDimet [LD92] and Kirk-Livingston-Wang [KW01].The unreduced representation could have been obtained with H = Hχ

1 (Xτ , I;Z[t±1]) where I is asegment reliying a base point of Dn to a base point of Dn′ .

3. Alexander polynomials

In this Section, we describe how the Alexander functors on the categories CobG and Tangµprovide models for the Alexander polynomial of knots and 3-manifolds. Similar results are provedin the more general setting of abelian Reidemeister torsion, see [8]. We also illustrate how theAlexander function can be used to construct a planar algebra, giving a combinatorial local modelfor the Alexander polynomial of tangle diagrams.

3.0.1. Knots in three-manifolds. In this Section, G is free abelian, finitely generated.Let K be an oriented knot in an oriented homology 3-sphere N . Recall that the Alexander

polynomial of K is defined, up to multiplication by a monomial ±tk for k ∈ Z, as

∆K := ∆(MK , ϕK) = ∆0HϕK1 (MK) ∈ Z[t±1]

where MK is the complement of an open tubular neighborhood of K in N and ϕK : H1(MK)→ Zis the isomorphism mapping an oriented meridian µ ⊂ ∂MK of K to t.


We make MK a morphism 1→ 0 in the category Cob by choosing a boundary-parametrizationm : F (1, 0) → ∂MK such that µ− := m−1(µ) is contained in the bottom surface F1 and goesthrough the base point ?. Set H− := H

ϕKm−1 (F1, ?). The following proposition shows that the knot

invariants ∆K and A(MK , ϕK) carry the same topological information.

Theorem 3.1. [8] With the above notation and for any h ∈ ΛiH−, we have

A(MK , ϕK)(h) =

∆K · ∂∗(h)/(t− 1) if i = 1,0 otherwise,

where ∂∗ : H− → Z[G] is the connecting homomorphism for the pair (F1, ?). In particular, we have

∆K = A(MK , ϕK)([µ−]).

Let N be a closed connected orientable 3-manifold, and let ϕ : H1(N) → G be a non-trivialgroup homomorphism, where G is free abelian. We wish to compute the Alexander polynomial ofN with coefficients ϕ, namely

∆(N,ϕ) = ∆0Hϕ1 (N) ∈ Z[G].

For this, we have to transform N into a cobordism. Note that removing an open 3-ball B fromN and regarding N \ B as an element of Cob(0, 0) is not fruitful, since the functor R maps thismorphism to zero.

We proceed in the following (rather indirect) way. Choose a knot K ⊂ N such that ϕ([K]) 6= 1.Consider the complement MK of an open tubular neighborhood of K in N , and fix a parallelρ ⊂ ∂MK of K. Let ϕK : H1(MK) → G be the homomorphism obtained from ϕ by restrictionto MK ⊂ N . Make MK a morphism 1 → 0 in Cob by choosing a boundary-parametrizationm : F (1, 0)→ ∂MK such that ρ− := m−1(ρ) is contained in the bottom surface F1 and ? ∈ ρ−.

Proposition 3.2. With the above notation, we have

∆(N,ϕ) =

A(MK , ϕK)([ρ−])

(ϕ([K])− 1)2if rankϕ(H1(N)) ≥ 2,

A(MK , ϕK)([ρ−])

(tn−1 + · · ·+ t+ 1)2if rankϕ(H1(N)) = 1.

In the second case, t ∈ ϕ(H1(N)) is a generator and n ∈ N is such that ϕ([K]) = tn.

3.0.2. Alexander and Jones polynomials of (1, 1)-Tangles. We study the special case of (1, 1)-tangles, corresponding to morphisms in Tang such that n− = n+ = 1 (ε− and ε+ have length 1).For a (1, 1)-tangle τ , we denote the link in S3 obtained by its closure τ .

Theorem 3.3 (7). Let (τ, ψ) : (ε−, ϕ−) → (ε+, ϕ+) be a (1, 1) colored tangle in Tangµ. Theimage by the Alexander functor (see Section 2.4) has the form

B(τ, ψ) : S → S,

and is given by the multiplication by the Alexander polynomial ∆(τ , ψ) ∈ S.

Remark 3.4. The functor B vanishes on closed tangles. To ’capture’ the Alexander polynomialof a link, we have to consider it as the closure of a (1, 1)-tangle, in the same way that we removeda full torus to capture the Alexander polynomial of a closed 3-manifold with the functor A, seeProposition 3.2.


Let (ε, ϕ) be an object in Tangµ. For all p ≥ 1, consider the symmetric product Sp(Dn) ofunordered p-tuples of points in Dn. Note that an element of π1(Sp(Dn)) is a singular braid inDn × [0; 1]. Let ν : π1(Sp(Dn)) → H be the morphism induced by the total winding numbers ofsuch a braid around the holes colored by ϕ, according to the signs of ε. The homology of S∗(Dn)twisted by ν coincides with the exterior algebra ΛHϕε

1 (Dn;S). The action of a colored braid on Dn

induces an automorphism of this homology, which coincides with the exterior powers of its imageby the Burau representation. In other terms, the functor B can be defined on the object with thehomology of symetric products of Dn and on the isomorphisms of Tangµ by their action on thishomology.

Let now (τ, ψ) be a (1, 1)-tangle, obtained as the plat closure of a braid σ ∈ B2n+1. Thehomology of S∗(Dn) can be endowed with a Hermitian structure coming from the exterior powerof the intersection form Hϕε

1 (Dn) × Hϕε1 (Dn) → S. In particular, the calculation of B applied to

this decomposition of (τ, ψ) as successive cups, σ and caps can be interpreted as the intersection oftwo-submanifolds in the symmetric power, in the spirit of Heedgaard-Floer homology. More detailscan be found in [Kal15]. This could be viewed as a topological interpretation of the description ofthe Alexander polynomial of a plat closure through the representations of Hecke algebras, by Jones(see [Jon87, Formula (7.2)]).

We note that a similar functor were constructed by R.Lawrence [Law93] for the Jones polyno-mial, in terms of twisted homology of configuration spaces. This was a first topological interpretationof the decomposition of the Ocneanu trace in terms of characters, see Chapter 3 Section 4.1. Thiswas used by S.Bigelow to describe the Jones polynomial of a plat closure as an intersection ofmanifolds in configuration spaces.

3.0.3. A Planar Algebra. We first define an invariant α of colored tangles, and then show thatα commute with the action of diagrams on tangles.

In this Section, we denote S = Z[t±11 , . . . , t±1

µ ] the Laurent polynomial ring. Let (τ, ψ) be aµ-colored welded tangle. It decomposes into a finite union of disjoint oriented arcs. Label thecrossings with (formal) letters, and each arc with the same letter as the crossing it begins at. Ifan arc connects points on the border of τ without meeting any crossing, we use the conventionof Figure 5. We construct a matrix Mψ(τ) with coefficients in S where the rows are indexed bycrossings (positive, negative and virtual) and points interrupting arcs, and the columns by the arcs.

• Fill row corresponding to each positive and negative crossing as shown in Figure 4,• At each point on the diagram, fill the row as shown in Figure 5.

The other entries of the rows are zero. Virtual crossings can be ignored or considered as dividedarcs. Notice that, after some Reidemeister moves of type I, one might suppose that there every arcbegins at a crossing, and the receipt of Figure 3 becomes useless to construct the matrix Mψ(τ).

Remark 3.5. Let p be the number of internal arcs of τ . Since τ has 2n arcs connected tothe boundary, the total number of arcs is 2n + p. One easily observes that the matrix Mψ(τ) hassize (p+ n)× (p+ 2n).

Definition 3.6. Let (τ, ψ) be a µ-colored welded n-tangle and H∂ be the S-module of rank 2nfreely generated by the set of marked points x1, . . . , x2n . The invariant α is defined to be

α(τ, ψ) =∑I

|Mψ(τ)I | · xI ∈ ΛnH∂ ,


c b

a

a b

c

tj

ti tj

c

c

−1

a

1

c

ti

1− ti

b

tj

c

−tj

a

ti − 1

b

Figure 4. The rule to fill the matrix Mψ(T ), where ti and tj are not necessarilydifferent. If b = a or b = c we add the contributions.

a

a

b

1

a

−1

b

Figure 5. Rule for arcs that don’t begin at crossings.

where the sum is taken for all subset I ⊂ 1, . . . , 2n of n elements, |Mψ(τ)I | is the determinantof the (n + p)-minor of Mψ(τ) corresponding to the columns indexed by the internal arcs and thecolumns relative to the arcs indexed by I, and xI is the wedge product of the generators xi with i ∈ I.

A computation shows that α(τ, ψ) is invariant by generalized Reidemeister moves, up to multi-plication by a unit.

Example 3.7. Consider the tangle τ given by one positive crossing. The matrix Mψ(τ) coincideswith the matrix of Example 2.7. The module H∂ is generated by x1, . . . , x4 and

α(τ, ψ) = t2x3 ∧ x4 + x2 ∧ x3 − t2x1 ∧ x4 + (t1 − 1)x1 ∧ x3 + x1 ∧ x2 ∈ Λ2H∂ .

We consider now the operad Dµ of µ-colored diagrams defined in Section 1.2.2. Let Homµ be theoperad of tensor products of S-modules and S-linear maps. The planar algebra Aµ is constructedas a morphism of operads from Dµ to Homµ.

Consider the unit circle with a base point and a set of marked points X = x1, . . . , x2k, for k ≥ 0(with a sign). To this data, we associate the module ΛkH∂ , where H∂ is the free S-module of rank2k generated by X. Let (P,ψ) be a µ-colored diagram, and M = c1, . . . , cq be the set of curvesof P . Consider the free module H generated by M , and the volume form ω : ΛqH → S related tothis basis. For i = 1, . . . , p, denote H∂i the module associated to the boundary circle ∂Di and H∂

he module associated to ∂D. Let mi : H∂i → H be the morphims defined by mi(xj) = sign(xj)cj ifxj ∈ ∂cj . The morphism m∂ is defined similarly. Let ω∂i be the volume form on H∂i related to thegenerating system of points of the circle ∂Di. Set m = ⊗i (Λnim∂i). To the colored diagram (P,ψ)we associate

γP,ψ :

p⊗i=1

ΛniH∂i → ΛnH∂


such that, for x ∈ ⊗i(ΛniH∂i

),

ωϕC(m(x) ∧m∂(y)) = ω∂(γP,ψ(x) ∧ y), ∀y ∈ ΛnH∂ .

Proposition 3.8. Aµ is a planar algebra.

Note that Aµ is similar to half densities introduced by Archibald, see [Arc] for the definition ofhalf densities and [Dam16] for the explicit correspondance. The morphism γP,ψ could be writtenas the interior product relative to a subset corresponding to interior arcs of P .

The invariant α commutes with this action of µ-colored diagrams on µ-colored tangles. Indeed,one has the following proposition.

Proposition 3.9. Let (τ, ψ) be the µ-colored tangle obtained by gluing the µ-colored tangles(τ1, ψ1), · · · , (τp, ψp) to a µ-colored diagram (P, χ). The following equality holds

α(τ, ψ) = γP,χ(α(τ1, ψ1)⊗ · · · ⊗ α(τp, ψp)

)∈ ΛnH∂ .

Example 3.10. Consider the tangle diagram τ , given in Figure 6. We let µ = 1 and ψ be thecoloring sending all arcs of τ to t. In this section, we compute α(τ, ψ) in different ways.

?

x1

x2

x3

x4x5

x6a

e

f

b

c

d

Figure 6. The labelled tangle τ .

First, we compute α(τ, ψ) directly. Label the arcs of τ with letters a to f as in Figure 6. Weobtain the matrix

Mψ(τ) =

a b c d e f

0 0 −1 1 0 0−1 0 0 0 1 0

1− t −1 0 0 0 t

The Z[t±1]-module H∂ is free, generated by x1, . . . , x6 and α(τ, ψ) ∈ Λ3H∂ is given by

α(τ, ψ) =− x1 ∧ x2 ∧ x5 + x1 ∧ x2 ∧ x6 + x2 ∧ x5 ∧ x4 + (t− 1)x1 ∧ x5 ∧ x4 − tx1 ∧ x5 ∧ x3

+(1− t)x1 ∧ x6 ∧ x4 + tx1 ∧ x6 ∧ x3 + tx6 ∧ x4 ∧ x3 − x2 ∧ x6 ∧ x4 − tx5 ∧ x4 ∧ x3.

We now consider (τ, ψ) as the composition of the diagram (P,ψ) with (σ, ψ), see Figure 7. We haveto compute γP,ψ : Λ2H∂1 → Λ3H∂ (here p = 1). Let H be the free module generated by the curvesof σ, labelled a, b, e, f . We have that H∂1 = 〈x1, . . . , x4〉 and H∂ = 〈x1, . . . , x6〉. Using the volumeform on H related to the choice of the basis a, b, c, e, d, f , and the maps induced by the inclusionsm1 : H∂1 → H and m∂ : H∂ → H, we obtain

γC,ψ(xi ∧ xj) = xi ∧ xj ∧ (x6 − x5), ∀ i, j = 1, . . . , 4,


x1

x2

x3

a

ef

b

x4

?

?

x5

x6

c

d?

Figure 7. A welded tangle σ and a diagram P .

and

Mψ(σ) =

( a b e f

−1 0 1 00 −1 1− t t

)We get

α(σ, ψ) = x1 ∧ x2 + (t− 1)x1 ∧ x4 − tx1 ∧ x3 + x2 ∧ x4 − tx3 ∧ x4.

The composition α(τ, ψ) = γP,ψ(α(σ, ψ)) gives the result. Finally, we consider τ as the compositionof the tangle σ ⊗ β with the diagram Q, see Figure 8. Here p = 2, and

?

c

d

x5

x6

??

?

x1

x2

x6

x5

x4

x3

?

x1 x2

x4x3

a b

ef

Figure 8. A diagram Q and two tangles σ and β.

γQ,ψ : Λ2H∂1 ⊗H∂2 → Λ3H∂ .

As previously α(σ, ψ) = x1 ∧ x2 + (t − 1)x1 ∧ x4 − tx1 ∧ x3 + x2 ∧ x4 − tx3 ∧ x4 ∈ Λ2H∂1 andγ(β) = x6 − x5 ∈ H∂2 . The composition α(τ, ψ) = γQ,ψ (α(σ, ψ)⊗ α(β, ψ)) gives the result again.

4. The Magnus functor

4.1. The Lagrangian category. The category of Lagrangian relations were introduced byCimasoni and Turaev in [CT05]. Let R be a commutative ring without zero-divisors, and letR → R, r 7→ r be an involutive ring homomorphism. A skew-Hermitian R-module is a finitelygenerated R-module H equipped with a non-degenerate skew-Hermitian form ρ : H × H → R.(In particular, the R-module H has no torsion.) Given a submodule A of H, one can consider itsannihilator with respect to ρ

Ann(A) := x ∈ H : ρ(x,A) = 0and its closure

cl(A) := x ∈ H : ∃r ∈ R \ 0, rx ∈ A.

4. THE MAGNUS FUNCTOR 77

A submodule A of H is said to be Lagrangian if A = Ann(A).We now recall some material from [CT05, Section 2.3]. Let (H1, ρ1) and (H2, ρ2) be skew-

Hermitian R-modules. A Lagrangian relation between (H1, ρ1) and (H2, ρ2) is a submodule N ofH1 ⊕H2 which is Lagrangian with respect to the skew-Hermitian form (−ρ1)⊕ ρ2; in this case, wedenote N : (H1, ρ1)⇒ (H2, ρ2). According to [CT05, Theorem 2.7], there is a category LagrR whoseobjects are skew-Hermitian R-modules, and whose morphisms are Lagrangian relations and arecomposed in the following way. The composition N2N1 of two Lagrangian relations N1 : (H1, ρ1)⇒(H2, ρ2) and N2 : (H2, ρ2)⇒ (H3, ρ3) is the closure cl(N2N1) of the following submodule of H1⊕H3:

(4.1) N2N1 :=

(h1, h3) ∈ H1 ⊕H3 : ∃h2 ∈ H2, (h1, h2) ∈ N1 and (h2, h3) ∈ N2

Note that, for any skew-Hermitian R-module (H, ρ), the diagonal

(h, h) ∈ H ⊕ H

∣∣h ∈ H is aLagrangian relation (H, ρ)⇒ (H, ρ), and constitutes the identity of the object (H, ρ) in the categoryLagrR.

We finally outline the relationship between Lagrangian submodules and graphs of unitary iso-morphisms following [CT05, Section 2.4]. Let Q := Q(R) be the field of fractions of R: there isa unique way to extend the involution r 7→ r of R to a ring homomorphism of Q. For any skew-Hermitian R-module (H, ρ), set HQ := Q ⊗R H: note that, since H is torsion-free, it embeds intoHQ by the map h 7→ 1⊗ h. Let ρ : HQ ×HQ → Q be the extension of ρ defined by

ρ(q ⊗ x, q′ ⊗ x′) := q q′ ρ(x, x′)

for any x, x′ ∈ H and q, q′ ∈ Q.A unitary Q-isomorphism (respectively, unitary R-isomorphism) between two skew-Hermitian

R-modules (H1, ρ1) and (H2, ρ2) is a Q-linear isomorphism ψ : (H1)Q → (H2)Q (respectively, aR-linear isomorphism ψ : H1 → H2) such that ρ2 (ψ × ψ) = ρ1. Let Ur

Q (respectively, UR) bethe category whose objects are skew-Hermitian R-modules and whose morphisms are unitary Q-isomorphisms (respectively, unitary R-isomorphisms). According to [CT05, Theorem 2.9], there areembeddings of categories UR → Ur

Q, UR → LagrR and UrQ → LagrR which fit in the commutative

diagram

UR UrQ

LagrR

and which are defined by ψ 7→ Q⊗R ψ, ψ 7→ Γψ and ψ 7→ Γrψ, respectively. Here Γψ denotes the

graph of a R-linear isomorphism ψ, while Γrψ denotes the restricted graph of a Q-linear isomorphism

ψ : (H1)Q → (H2)Q and is defined by

Γrψ :=

(h1, h2) ∈ H1 ⊕H2 : h2 = ψ(h1)

.

Let R be a commutative ring without zero-divisors, and let R→ R, r 7→ r be an involutive ringhomomorphism. We introduce a refinement of the category LagrR, which seems to be new.

A pointed skew-Hermitian R-module is a skew-Hermitian R-module (H, ρ) equipped with adistinguished element s ∈ HQ satisfying ρ(s, s) = 0 and ρ(s,H) ⊂ R, where ρ : HQ × HQ → Qdenotes here the extension of ρ to HQ = Q⊗RH. Let (H1, ρ1, s1) and (H2, ρ2, s2) be pointed skew-Hermitian modules: a pointed Lagrangian relation N : (H1, ρ1, s1) ⇒ (H2, ρ2, s2) is a Lagrangian


submodule N of (H1 ⊕H2, (−ρ1)⊕ ρ2) such that

(s1, s2) ∈ NQ := Q⊗R N ⊂ (H1)Q ⊕ (H2)Q.

The composition of Lagrangian relations induces a composition rule for pointed Lagrangian relations.Thus we get the category pLagrR of pointed Lagrangian relations. Similarly, we can define somerefinements pUR and pUr

Q of the categories UR and UrQ, respectively, by requiring that (the extensions

of) the unitary R-isomorphisms and the unitary Q-isomorphims preserve the distinguished elements.All these categories fit together into the commutative diagram

UR UrQ

pUR pUrQ Lagr

pLagr

where the arrows pLagrR → LagrR, pUR → UR, pUrQ → Ur

Q denote the forgetful functors.We now enrich the category pLagrR with a monoidal structure. For any pointed skew-Hermitian

modules (H, ρ, s) and (H ′, ρ′, s′), let ρ s⊕s′ρ′ : (H ⊕H ′)× (H ⊕H ′)→ R be defined by

(ρ s⊕s′ρ′)(h1 + h′1, h2 + h′2) := ρ(h1, h2) + ρ′(h′1, h′2) + ρ(h1, s) ρ

′(s′, h′2)− ρ(s, h2) ρ′(h′1, s′)

for any h1, h2 ∈ H and h′1, h′2 ∈ H ′.

Proposition 4.1. There is a monoidal structure on the category pLagrR defined by

(4.2) (H, ρ, s)⊗ (H ′, ρ′, s′) := (H ⊕H ′, ρ s⊕s′ρ′, s+ s′)

for any objects (H, ρ, s), (H ′, ρ′, s′) in pLagrR, and by

(4.3) N ⊗N ′ := N ⊕N ′

for any morphisms N : (H, ρ, s)⇒ (K, τ, t) and N ′ : (H ′, ρ′, s′)⇒ (K ′, τ ′, t′) in pLagrR.

Formally speaking, the monoidal category pLagrR is not strict. But, since its associativity andunit constraints arise from canonical bijections in set theory, we will assume in the sequel thatpLagrR is strict monoidal. Moreover, the category LagrR itself has a strict monoidal structure,which is defined by (H, ρ) ⊗ (H ′, ρ′) := (H ⊕H ′, ρ ⊕ ρ′) at the level of objects and by N ⊗ N ′ :=N ⊕ N ′ at the level of morphisms. The embedding of categories LagrR → pLagrR that is definedby (H, ρ) 7→ (H, ρ, 0) at the level of objects, and that is the identity at the level of morphisms, isstrictly monoidal.

4.2. Intersection forms. Let g ≥ 0 be an integer and consider the compact, connected, ori-ented surface Fg of genus g with one boundary component. In this section, we review Turaev’s“homotopy intersection pairing” on Fg, which is a non-commutative version of Reidemeister’s “equi-variant intersection forms”. An advantage of the former with respect to the latter is a simplerdefinition, avoiding the use of covering spaces and allowing for a straightforward verification of themain properties.

4. THE MAGNUS FUNCTOR 79

4.2.1. The homotopy intersection pairing λ. Set π := π1(Fg, ?) where ? ∈ ∂Fg. We denoteby Z[π] the group ring of π and we denote by a 7→ a the antipode of Z[π], which is the anti-homomorphism of rings defined by x = x−1 for any x ∈ π. The homotopy intersection pairing of Fgis the pairing

λ : Z[π]× Z[π] −→ Z[π]

introduced by Turaev in [Tur78]. Recall that the map λ is Z-bilinear and λ(a, b) ∈ Z[π] is definedas follows for any a, b ∈ π. Let ν be the oriented boundary curve of Fg. Let l, r ∈ ∂Fg be someadditional points such that l < ? < r along ν. Given an oriented path γ in Fg and two simple pointsp < q along γ, we denote by γpq the arc in γ connecting p to q, while the same arc with the oppositeorientation is denoted by γqp. Let α be a loop based at l such that ν?lανl? represents a and let βbe a loop based at r such that ν?rβνr? represents b; we assume that these loops are in transverseposition and that α ∩ β only consists of simple points of α and β:

l r

α β

p

Fg

?

ν

Then

(4.4) λ(a, b) :=∑p∈α∩β

εp(α, β) ν?lαlpβprνr?

where the sign εp(α, β) = ±1 is equal to +1 if, and only if, a unit tangent vector of α followed bya unit tangent vector of β gives a positively-oriented frame of the oriented surface Fg. The pairingλ is implicit in Papakyriakopoulos’ work [Pap75]; see also [Per06].

Note that λ(Z1,Z[π]) = λ(Z[π],Z1) = 0: hence λ is determined by its restriction to the aug-mentation ideal I(Z[π]) of the group ring Z[π]. This restriction has several properties observed byTuraev in [Tur78]. See also [12, Section 3.1] Since π is a free group of rank 2g, I(Z[π]) is a freeleft Z[π]-module of rank 2g. Furthermore, I(Z[π]) can be identified with H1(Fg, ? ; Z[π]) using theconnecting homomorphism

∂∗ : H1(Fg, ? ; Z[π]) −→ H0(?;Z[π]) ' Z[π]

in the long exact sequence of the pair (Fg, ?).

4.2.2. The twisted intersection form 〈·, ·〉s. Assume now the following:

(4.5)

R is a commutative ring without zero-divisors such that 2 6= 0 ∈ R;G ⊂ R× is a multiplicative subgroup of the group of units of R;R has an involutive ring endomorphism r 7→ r satisfying x = x−1 for all x ∈ G.

Any group homomorphism ϕ : H1(Fg) → G induces a ring homomorphism ϕ : Z[π] → R, whichgives R the structure of a right Z[π]-module. Thus we can consider the twisted homology groupHϕ

1 (Fg, ?) which, as an R-module, can be identified with

R⊗Z[π] H1(Fg, ? ; Z[π]) ' R⊗Z[π] I(Z[π]).


Using this identification, we define a pairing 〈·, ·〉 : Hϕ1 (Fg, ?)×Hϕ

1 (Fg, ?)→ R by setting

(4.6) ∀r, r′ ∈ R, ∀x, x′ ∈ I(Z[π]), 〈r ⊗ x, r′ ⊗ x′〉 := rr′ ϕ(λ(x, x′)

).

This pairing is well-defined and sesquilinear; but it is not quite skew-symmetric : 〈x, y〉 = −〈y, x〉+∂∗(x) ∂∗(y) for any x, y ∈ Hϕ

1 (Fg, ?). Therefore, we will prefer to 〈·, ·〉 the skew-Hermitian form

〈·, ·〉s : Hϕ1 (Fg, ?)×Hϕ

1 (Fg, ?) −→ R

defined by

(4.7) 〈x, y〉s := 2〈x, y〉 − ∂∗(x) ∂∗(y)

and which will be referred to as the ϕ-twisted intersection form of Fg. A matrix presentation of 〈·, ·〉scan be obtained in a system of meridians and parrallels, through straightforward computations. Inparticular, it can be verified that the determinant of this matrix Sϕ is 4g: therefore the form 〈·, ·〉sis non-degenerate. Moreover,

(4.8) ∀x ∈ Hϕ1 (Fg, ?), 〈x, ν〉s = 2 ∂∗(x).

We deduce that 〈ν, ν〉s = 0, so that(Hϕ

1 (Fg, ?), 〈·, ·〉s, ν/2)

is a pointed skew-Hermitian R-modulein the sense of Section 4.1.

4.2.3. The equivariant intersection form S. We now briefly recall a few facts about Reidemeis-ter’s equivariant intersection forms [Rei39]. Let N be a piecewise-linear compact connected orientedn-manifold, and let J, J ′ be two disjoint subsets of ∂N . (We possibly have J = ∅ or J ′ = ∅, oreven ∂N = ∅.) We assume that N is endowed with a triangulation T , such that J is a subcomplexof T and J ′ is a subcomplex of the dual cellular decomposition T ∗.

Fix a ring R and a multiplicative subgroup G ⊂ R× as in (4.5), and let ϕ : H1(N) → G be agroup homomorphism. The equivariant intersection form of N (with coefficients in R twisted by ϕ,relative to J t J ′) is a sesquilinear map

SN := SN,ϕ,JtJ ′ : Hϕq (N, J)×Hϕ

n−q(N, J′) −→ R.

Blanchfield’s duality theorem adapted to our setting can be easily deduced from [Bla57, Theorem2.6]:

Theorem 4.2 (Blanchfield). The left and right annihilators of SN are the torsion submodulesof Hϕ

q (N, J) and Hϕn−q(N, J

′), respectively.

Assume now that N := Fg. Let l, r ∈ ∂Fg be some points such that l < ? < r if we follow∂Fg in the positive direction. Then the arc joining l to ? in ∂Fg \ r induces an isomorphismHϕ

1 (Fg, ?)'Hϕ1 (Fg, l) and, similarly, the arc joining ? to r in ∂Fg \ l induces an isomorphism

Hϕ1 (Fg, ?)'Hϕ

1 (Fg, r). It is easily deduced from the definitions that the diagram

(4.9) Hϕ1 (Fg, l)×Hϕ

1 (Fg, r)S // R

Hϕ1 (Fg, ?)×Hϕ

1 (Fg, ?) 〈·,·〉//

'OO

R

is commutative, where we denote S := SFg ,ϕ,ltr and 〈·, ·〉 is the pairing defined by (4.6). Sincethe form S is non-degenerate by Theorem 4.2, we recover from (4.7) and (4.8) the fact that 〈·, ·〉s isnon-degenerate.

5. A FIRST STEP IN DIMENSION 4 81

4.3. Definition of Mag.

Theorem 4.3. [12] There is a functor Mag := MagR,G : CobG → pLagrR defined by

Mag(g, ϕ) :=(Hϕ

1 (Fg, ?), 〈·, ·〉s, ν/2)

for any object (g, ϕ) of CobG, and by

Mag(M,ϕ) := cl(

ker((−m−)⊕m+ : H

ϕ−1 (Fg− , ?)⊕Hϕ+

1 (Fg+ , ?)→ Hϕ1 (M, ?)

))for any morphism (M,ϕ) : (g−, ϕ−)→ (g+, ϕ+) in CobG.

This is an analogue of [CT05, Theorem 3.5] and [CT05, Theorem 6.1] for cobordisms.We show two fundamental properties of the Magnus functor. The first one involves the following

relation among cobordisms. Two cobordisms (M1, ϕ1), (M2, ϕ2) ∈ CobG((g−, ϕ−), (g+, ϕ+)) arehomology concordant if there exists a compact connected oriented 4-manifold W with

∂W = M1 ∪m1m−12

(−M2)

such that the inclusion maps i1 : M1 → W and i2 : M2 → W induce isomorphisms at the level ofH(·;Z) and satisfy ϕ1(i1)−1 = ϕ2(i2)−1 : H1(W )→ G. In such a situation, we write (M1, ϕ1) ∼H(M2, ϕ2). It is easily verified that ∼H is an equivalence relation on the set CobG((g−, ϕ−), (g+, ϕ+))for any objects (g−, ϕ−), (g+, ϕ+), and that ∼H defines a congruence relation on the category CobG.Besides, the monoidal structure of CobG induces a monoidal structure on the quotient categoryCobG/∼H .

In addition to the hypothesis (4.5) on the ring R and the multiplicative subgroup G ⊂ R×,consider the following condition:

(4.10) R is equipped with a ring homomorphism εR : R→ Z such that εR(G) = 1.

Proposition 4.4. Under the assumption (4.10), the functor Mag : CobG → pLagrR descends tothe quotient CobG/∼H .

Proposition 4.4 is the analogue of the invariance under concordance of tangles which Cimasoniand Turaev mentioned for their functor [CT05, end of §3.3].

Lemma 4.5. Assume (4.10). Let (X,Y ) be a pair of CW-complexes such that the inclusionmap i : Y → X induces an isomorphism at the level of H(· ;Z). For any group homomorphismϕ : H1(X)→ G, we have HϕQ(X,Y ) = 0 where ϕQ : Z[H1(X)]→ Q(R) is the ring homomorphisminduced by ϕ.

The second property of the Magnus functor to be shown is the monoidality. The tangle analogueof Proposition 4.6 does not seem to have been addressed in [CT05].

Proposition 4.6. The functor Mag : CobG → pLagrR is strongly monoidal.

5. A first step in dimension 4

In this section G is a free abelian group, and S is the group ring Z[G].


5.1. Ribbon 2-links and tangles. Let m be a positive integer and X a submanifold of the m-dimensional ball Bm. An immersion Y ⊂ X is locally flat if and only if it is locally homeomorphicto a linear subspace Rk in Rm for some k ≤ m, except on ∂X and/or ∂Y , where one of the Rsummands should be replaced by R+. An intersection Y1 ∩Y2 ⊂ X is flatly transverse if it is locallyhomeomorphic to the intersection of two linear subspaces Rk1 and Rk2 in Rm for some positiveintegers k1, k2 ≤ m except on ∂X, ∂Y1 and/or ∂Y2, where one of the R summands is replacedby R+.

An intersection D = Y1 ∩ Y2 ⊂ S4 is a ribbon disk if it is homeomorphic to the 2-disk andsatisfies: D ⊂ Y1, D ⊂ Y2 and ∂D is an essential curve in ∂Y2. More details on ribbon knottedobjects can be found in [ABMW14, Aud15].

Definition 5.1. Let L be an oriented trivial link with 2n components in S3 = ∂B4. A ribbontangle T is a locally flat proper embedding in B4 of oriented disjoint annuli S1×I denoted A1, . . . , Anand disjoint tori S1 × S1 denoted E1, . . . , Em such that:

• There exist locally flat immersed solid tori Fi for i = 1, . . . ,m such that ∂Fi = Ei.• ∂Ai ⊂ L and the orientation induced by Ai on ∂Ai coincides with the given orientation of

the two components of L.• There exists n locally flat immersed 3-balls Bi ' B2×I whose singular set are finite number

of ribbon disks and such that, for all i ∈ 1, . . . , n:∂Bi = Ai ∪∂ (B2 × 0, 1).

A G-colored ribbon tangle is a pair (T, ϕ) where T is a ribbon tangle with complement XT =B4 \ T , equipped with a group homomorphism ϕ : H1(XT )→ G.

5.2. Definition of α. Let (T, ϕ) be a G-colored ribbon tangle. For the rest of this section, weset H = Hϕ

1 (XT , ∗;S).

Proposition 5.2. The S-module H admits a presentation with deficiency n.

Let H∂ = Hϕ1 (S3 \ L, ∗) be the free S-module of rank 2n, generated by the meridians of L. Let

m∂ : H∂ → H be induced by the inclusion map S3 \ L → XT . For short, for a given z ∈ ΛnH∂ , weuse the notation m∂z for ∧nm∂(z).

Definition 5.3. The element α(T, ϕ) of ΛnH∂ is the (colored) isotopy invariant defined by thefollowing property:

(5.1) ∀z ∈ ΛnH∂ , AϕT (m∂z) = ω∂(α(T, ϕ) ∧ z)where ω∂ is a volume form on H∂ .

Definition 5.4. The Alexander polynomial of a G-colored ribbon tangle (T, ϕ) is

∆(T, ϕ) := ∆0 (Hϕ1 (XT )) ∈ R.

Similarly to classical knot theory, T is a (1, 1)-tangle if n = 1. The components of T in B4 consistof m tori and a cylinder whose boundary is a 2-component trivial link L in S3 = ∂B4. Let x1 andx2 be the meridians of the components of L. Note that in XT , both x1 and −x2 are homologous tothe meridian x of the cylinder. We use the same notations x1 and x2 for the homology classes oftheir lifts in Hϕ

1 (S3 \ L, ∗).

5. A FIRST STEP IN DIMENSION 4 83

Proposition 5.5. Let (T, ϕ) be a G-colored (1−1)-tangle, such that ϕ is not trivial. Denote t =ϕ(x) and let r be the rank of ϕ(H1(XT )). Then the element α(T, ϕ) of H∂ is given by

α(T, ϕ) =

(t− 1)∆(T, ϕ) · (x1 − x2) if r ≥ 2,

∆(T, ϕ) · (x1 − x2) if r = 1.

It is worth noticing that, up to a unit in S, the result is independent of the order chosen on thecomponents of L.

5.3. The operad of colored cobordisms. In this section, we introduce the circuit alge-bra CobG. The algebraic structure is inspired by [Jon99, Pol10], see also [Ken13, Arc].

Definition 5.6. Let B = B0 be a 4-ball and B1, . . . , Bp be disjoint 4-balls in the interior of B.For every i ∈ 0, . . . , p, let Li (with L = L0) be a trivial oriented link with 2ni (n = n0) componentsin S3

i = ∂Bi (with S3 = S30). A cobordism C is a disjoint union of locally flat proper embedded

annuli in B \ B1, . . . Bp, whose boundary are the links Li, with the conditions of Definition 5.1but without singularities.

Definition 5.7. Let C ′ and C ′′ be two cobordisms such that B′i is a ball of C ′ with n′i = n′′.The composition C ′ i C ′′ is the cobordism obtained with the identification of B′′ = B′′0 with B′i.

As in the previous section, G is a fixed free abelian group with group ring R. A G-coloredcobordism is a pair (C,ϕ) where C is a cobordism with complement XC , equipped with is a grouphomomorphism ϕ : H1(B4\C)→ G. The orientation-preserving diffeomorphism classes of G-coloredcobordisms with composition of compatible cobordisms, form an operad denoted CG.

Let HomG be the operad of tensor powers of R-modules and R-linear applications, consideredup to an element of ±G. The composition in HomG is induced by the usual composition of maps.We construct the circuit algebra RibG as a morphism of operads from CG to HomG.

Let (L,ϕ) be a G-colored oriented trivial link with k components in S3, with complementXL = S3 \L. The group homomorphism ϕ : H1(S3 \L)→ G induces a ring homomorphism denotedalso ϕ : Z[H1(S3 \ L)]→ R. Let ∗ be a base point on S3. The R-module Hϕ

1 (S3 \ L, ∗;R) is free ofrank k, generated by the meridians of L.

Let now (C,ϕ) be a G-colored cobordism, with complement XC . For i = 1, . . . , p, let ∗ and ∗i bebase points in the boundary of Bi and Ji be intervals (whose interiors are disjoint, and disjoint fromC) connecting ∗ to ∗i. Note that the union of the Ji is contractible. The homomorphism ϕ inducesa ring homomorphism Z[H1(XC)] → R also denoted ϕ. The inclusion mi : S

3i \ Li → XT induces

ϕi : Z[H1(S3i \ Li)] → R. Set H = Hϕ

1 (XC , J), H∂ = Hϕ1 (S3 \ L, ∗), and H∂i = Hϕi

1 (S3 \ Li, ∗i)for i = 1, . . . , p. Let ωϕC be a volume form ωϕC : ΛrH → R, and ω∂ : Λ2nH∂ → R. For i = 1, . . . , p, we

denote again mi : H∂i → H the map induced by the inclusion. Let m : ⊗i ΛniH∂i → Λn1+···+npH bedefined as m = Λn1m1 ∧ · · · ∧ Λnpmp.

To the cobordism (C,ϕ) we associate

ΥC,ϕ :

p⊗i=1

ΛniH∂i → ΛnH∂

such that, for x ∈ ⊗i(ΛniH∂i

),

(5.2) ωϕC(m(x) ∧m∂(y)) = ω∂(ΥC,ϕ(x) ∧ y), ∀y ∈ ΛnH∂ .


We denote RibG the object that associates to a pair (L,ϕ) the homology module Hϕ1 (S3\L, ∗;R),

and to each pair (C,ϕ) the linear map ΥC,ϕ.

Theorem 5.8. [13] RibG is a planar algebra.

5.3.1. Action of cobordisms on tangles. Given a cobordism C and a collection of tangles T1, · · · , Tp,one may create a new tangle, if n(Ti) = ni for all i = 1, . . . , p, by gluing each Ti into the internalball Bi of C. The action of G-colored cobordisms on G-colored tangles is defined once the coloringscoincide on the boundary components. The following theorem states that the invariant A respectsthe structure of planar algebra RibG.

Theorem 5.9. [13] Let (T, ψ) be the G-colored tangle obtained by gluing the G-colored tangles(T1, ϕ1), · · · , (Tp, ϕp) to a G-colored cobordism (C,ϕ). The following equality holds

A(T, ψ) = ΥC,ϕ

(A(T1, ϕ1)⊗ · · · ⊗ A(Tp, ϕp)

)∈ ΛnH∂ .

6. Perspectives

6.1. Twisted Alexander polynomials. This project is a part of the suject of the PHD thesisof J.Serrano. This part is advised by G.Massueyau and me. The aim is to extend the Reidemeisterand Alexander functors (see Section 2) to non-abelian settings. The coefficients are twisted by alinear representation, similarly to Section 2.1.

Problem 6.1. Let G be a free abelian group, and V be a F-vector space of finite dimension.Define a category CobG,V of 3-dimensional cobordisms endowed with a G-representation and aGL(V )-representation. Construct a functor from CobG,V to the category of F[t±1]-modules such thatthe image of a knot complement viewed as a morphism 0 → 1 is the twisted torsion or Alexanderpolynomial of the knot (see Theorem 3.1 for the abelian case).

6.2. The Alexander and Jones polynomials. This follows Section 3.0.2.

Question 6.2. Let τ be a tangle with exterior X, and Alexander module H1(X;Z[t±11 , . . . , t±1

µ ]).Does there exist a topological interpretation of the Alexander function (see Definition 2.6)

A : ΛkH1(X;Z[t±11 , . . . , t±1

µ ])→ Z[t±11 , . . . , t±1

µ ]

in terms of the twisted homology of the symetric power S∗(X)?

The same interpretation of the Alexander functor B in terms of symmetric powers (or configu-ration spaces) could be written for the functor A on CobG. The homology of symmetric powers ofhigher genus surfaces with one boundary component is discussed in [KS06].

Question 6.3. Does there exists a function similar to the Alexander function, defined on thetwisted homology of the configuration spaces of the exterior of a tangle, giving a topological interpre-tation of Lawrence’s functor for the Jones polynomial?

Bibliography

[AB14] E. Artal Bartolo, Topology of arrangements and position of singularities, Ann. Fac. Sci. Toulouse Math.(6) 23 (2014), no. 2, 223–265. MR 3205593

[ABCT06] Enrique Artal Bartolo, Jose Ignacio Cogolludo, and Hiro-o Tokunaga, Nodal degenerations of plane curvesand Galois covers, Geom. Dedicata 121 (2006), 129–142. MR 2276239

[ABCT08] E. Artal Bartolo, J. I. Cogolludo, and H. Tokunaga, A survey on Zariski pairs, Algebraic geometry inEast Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100.MR 2409555

[ABMW14] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner, Homotopy classification of ribbon tubes andwelded string links, ArXiv e-prints (2014).

[ACCAM05] E. Artal, J. Carmona, J.I. Cogolludo-Agustın, and M.A. Marco, Topology and combinatorics of real linearrangements, Compos. Math. 141 (2005), no. 6, 1578–1588.

[ACCAM07] , Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory andits applications (Tokyo) (S. Izumiya, G. Ishikawa, H. Tokunaga, I. Shimada, and T. Sano, eds.), AdvancedStudies in Pure Mathematics, vol. 43, Mathematical Society of Japan, 2007.

[ACCT01] E. Artal, J. Carmona, J. I. Cogolludo, and Hiro-O Tokunaga, Sextics with singular points in specialposition, J. Knot Theory Ramifications 10 (2001), no. 4, 547–578. MR 1831676

[Ale28] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2,275–306. MR 1501429

[APS75] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. II, Math.Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 405–432. MR 0397798

[Ara97] Donu Arapura, Geometry of cohomology support loci for local systems. I, J. Algebraic Geom. 6 (1997),no. 3, 563–597. MR 1487227

[Arc] J. Archibald, The multivariable alexander polynomial on tangles, Ph.D. thesis, Graduate Department ofMathematics, University of Toronto.

[Arn71] V. I. Arnol′d, The situation of ovals of real plane algebraic curves, the involutions of four-dimensionalsmooth manifolds, and the arithmetic of integral quadratic forms, Funkcional. Anal. i Prilozen. 5 (1971),no. 3, 1–9. MR 0286790

[Art25a] E. Artin, Theorie der Zopfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), no. 1, 47–72. MR 3069440[Art25b] , Zur Isotopie zweidimensionaler Flachen im R4, Abh. Math. Sem. Univ. Hamburg 4 (1925),

no. 1, 174–177. MR 3069446[Art94] E. Artal, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223–247. MR 1257321

(94m:14033)[Art14] E. Artal, Topology of arrangements and position of singularities, Ann. Fac. Sci. Toulouse Math. (6) 23

(2014), no. 2, 223–265.[Arv92] W.A. Arvola, The fundamental group of the complement of an arrangement of complex hyperplanes,

Topology 31 (1992), 757–765.[AS68] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604.

MR 0236952 (38 #5245)[Ati88] M Atiyah, Topological quantum field theories, Publications mathmatiques de l’I.H..S. 68 (1988), 175 –

186.[Aud15] B. Audoux, On the welded tube map, (to appear in) Contemporary Math. (2015).

85

86 BIBLIOGRAPHY

[BCGPM16] C. Blanchet, F. Costantino, N. Geer, and B. Patureau-Mirand, Non-semi-simple TQFTs, Reidemeistertorsion and Kashaev’s invariants, Adv. Math. 301 (2016), 1–78. MR 3539369

[Bes05] D. Bessis, Variations on Van Kampen’s method, J. Math. Sci. (N. Y.) 128 (2005), no. 4, 3142–3150,Geometry. MR 2171593 (2006g:55014)

[Bir74] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.;University of Tokyo Press, Tokyo, 1974, Annals of Mathematics Studies, No. 82.

[Bla57] R. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, TheAnnals of Mathematics 65 (1957), no. 2, 340–356.

[BLVS+99a] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented matroids, second ed.,Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge,1999. MR 1744046

[BLVS+99b] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler, Oriented matroids, second ed.,Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge,1999.

[Car03] J. Carmona, Monodromia de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza(2003).

[CG78] A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology(Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math.,XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521 (81g:57003)

[CG86] , Cobordism of classical knots, A la recherche de la topologie perdue, Progr. Math., vol. 62,Birkhauser Boston, Boston, MA, 1986, With an appendix by P. M. Gilmer, pp. 181–199. MR 900252

[Cha10] J. C. Cha, Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers, J.Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 555–610. MR 2639313

[Che73] D. Cheniot, Une demonstration du theoreme de Zariski sur les sections hyperplanes d’une hypersurfaceprojective et du theoreme de Van Kampen sur le groupe fondamental du complementaire d’une courbeprojective plane, Compositio Math. 27 (1973), 141–158. MR 0366922

[Cim04] D. Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv. 79(2004), no. 1, 124–146. MR 2031702

[Cim05] , The Conway function of a splice, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 1, 61–73.MR 2117712 (2005j:57007)

[Coc85] Tim D. Cochran, Geometric invariants of link cobordism, Comment. Math. Helv. 60 (1985), no. 2, 291–311. MR 800009

[Coo82] D. Cooper, The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979), LondonMath. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 51–66.MR 662427

[CS69] R. H. Crowell and D. Strauss, On the elementary ideals of link modules, Trans. Amer. Math. Soc. 142(1969), 93–109. MR 0247625

[CS97] D. C. Cohen and A. I. Suciu, The braid monodromy of plane algebraic curves and hyperplane arrange-ments, Comment. Math. Helv. 72 (1997), no. 2, 285–315. MR 1470093

[CS99a] , Alexander invariants of complex hyperplane arrangements, Trans. Amer. Math. Soc. 351 (1999),no. 10, 4043–4067. MR 1475679

[CS99b] D.C. Cohen and A.I. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Philos.Soc. 127 (1999), no. 1, 33–53.

[CS06] D.l C. Cohen and A. I. Suciu, Boundary manifolds of projective hypersurfaces, Adv. Math. 206 (2006),no. 2, 538–566. MR 2263714

[CS08] D.C. Cohen and A.I. Suciu, The boundary manifold of a complex line arrangement, Groups, homotopy andconfiguration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 105–146.

[CST16] J. Conant, R. Schneiderman, and P. Teichner, Covering linkage invariants in abelian coverings of links,arXiv:1607.01722 (2016).

[CSW14] J. S. Carter, D.l S. Silver, and S. G. Williams, Three dimensions of knot coloring, Amer. Math. Monthly121 (2014), no. 6, 506–514. MR 3225459

[CT05] D. Cimasoni and V. Turaev, A Lagrangian representation of tangles, Topology 44 (2005), no. 4, 747–767.[CT06] , A Lagrangian representation of tangles. II, Fund. Math. 190 (2006), 11–27.

BIBLIOGRAPHY 87

[CY99] L. Crane and D Yetter, On algebraic structures implicit in topological quantum field theories, Journal ofKnot Theory and its Ramifications 8 (1999), no. 2, 125–163.

[Dam16] C.. Damiani, The topology of loop braid groups: applications and remarkable quotients, Phd Thesis,Universite de Caen (2016).

[Dav06] J. F. Davis, A two component link with Alexander polynomial one is concordant to the Hopf link, Math.Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 265–268. MR 2212279

[Deg94] A. Degtyarev, Alexander polynomial of a curve of degree six, J. Knot Theory Ramifications 3 (1994),no. 4, 439–454. MR 1304394

[Deh10] M. Dehn, Uber die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168.MR 1511580

[Deh14] , Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), no. 3, 402–413. MR 1511799[Dur83] A.H. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983), no. 2, 517–530.[Dyc82] W. Dyck, Gruppentheoretische Studien, Math. Ann. 20 (1882), no. 1, 1–44. MR 1510147[EN85] D. Eisenbud and W. Neumann, Three-dimensional link theory and invariants of plane curve singularities,

Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982(87g:57007)

[Fel16] Peter Feller, The degree of the Alexander polynomial is an upper bound for the topological slice genus,Geom. Topol. 20 (2016), no. 3, 1763–1771. MR 3523068

[FM66] R. H. Fox and J. W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math.3 (1966), 257–267. MR 0211392

[FN62] R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126. MR 0150755[FN91] C. Frohman and A. Nicas, The Alexander polynomial via topological quantum field theory, Differential

geometry, global analysis, and topology (Halifax, NS, 1990), CMS Conf. Proc., vol. 12, Amer. Math.Soc., Providence, RI, 1991, pp. 27–40.

[FN94] , An intersection homology invariant for knots in a rational homology 3-sphere, Topology 33(1994), no. 1, 123–158.

[Fox62a] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ.of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099

[Fox62b] , Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. ofGeorgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 168–176. MR 0140100

[Fox70] , Metacyclic invariants of knots and links, Canad. J. Math. 22 (1970), 193–201. MR 0261584[GB16] B. Guerville-Balle, An arithmetic Zariski 4-tuple of twelve lines, Geom. Topol. 20 (2016), no. 1, 537–553.

MR 3470721[Gil81] P. M. Gilmer, Configurations of surfaces in 4-manifolds, Trans. Amer. Math. Soc. 264 (1981), no. 2,

353–380. MR 603768 (83h:57027)[Gil82] , On the slice genus of knots, Invent. Math. 66 (1982), no. 2, 191–197. MR 656619[GL92] P. Gilmer and C. Livingston, Discriminants of Casson-Gordon invariants, Math. Proc. Cambridge Philos.

Soc. 112 (1992), no. 1, 127–139. MR 1162937[GLM81] C. McA. Gordon, R. A. Litherland, and K. Murasugi, Signatures of covering links, Canad. J. Math. 33

(1981), no. 2, 381–394. MR 617628 (83a:57006)[Gue13] B. Guerville-Balle, Topological invariants of line arrangements, Ph.D. thesis, Universite de Pau et des

Pays de l’Adour and Universidad de Zaragoza, 2013.[Hab06] K. Habiro, Bottom tangles and universal invariants, Algebr. Geom. Topol. 6 (2006), 1113–1214 (elec-

tronic).[Hil81] Jonathan A. Hillman, Alexander ideals of links, Lecture Notes in Mathematics, vol. 895, Springer-Verlag,

Berlin-New York, 1981. MR 653808[Hir01] E. Hironaka, Boundary manifolds of line arrangements, Mathematische Annalen 319 (2001), 17–32.[Hur91] A. Hurwitz, Ueber Riemann’sche Flachen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891),

no. 1, 1–60. MR 1510692[Jon87] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2)

126 (1987), no. 2, 335–388. MR 908150[Jon99] V. F. R. Jones, Planar algebras, I, ArXiv Mathematics e-prints (1999).

88 BIBLIOGRAPHY

[Kal15] N. Kalinin, The Alexander polynomial as an intersection of two cycles in a symmetric power, J. KnotTheory Ramifications 24 (2015), no. 12, 1550061, 22. MR 3415241

[Ken13] K. G. Kennedy, A Diagrammatic Multivariate Alexander Invariant of Tangles, ProQuest LLC, AnnArbor, MI, 2013, Thesis (Ph.D.)–University of California, Santa Barbara. MR 3192920

[Ker03] T. Kerler, Homology TQFT’s and the Alexander-Reidemeister invariant of 3-manifolds via Hopf algebrasand skein theory, Canad. J. Math. 55 (2003), no. 4, 766–821.

[Kit96] T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), no. 2,431–442. MR 1405595

[KL99] P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordoninvariants, Topology 38 (1999), no. 3, 635–661. MR 1670420

[KS06] S. Kallel and P. Salvatore, Symmetric products of two dimensional complexes, Recent developments inalgebraic topology, Contemp. Math., vol. 407, Amer. Math. Soc., Providence, RI, 2006, pp. 147–161.MR 2248976

[KT76] L. H. Kauffman and L. R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351–365.MR 0388373

[KT00] V.. S. Kulikov and M. Taıkher, Braid monodromy factorizations and diffeomorphism types, Izv. Ross.Akad. Nauk Ser. Mat. 64 (2000), no. 2, 89–120. MR 1770673

[KW01] C. Kirk, P.and Livingston and Z. Wang, The Gassner representation for string links, Commun. Contemp.Math. 3 (2001), no. 1, 87–136.

[KY79] S. Kojima and M. Yamasaki, Some new invariants of links, Invent. Math. 54 (1979), no. 3, 213–228.MR 553219

[Law93] R. J. Lawrence, A functorial approach to the one-variable Jones polynomial, J. Differential Geom. 37(1993), no. 3, 689–710. MR 1217166

[LD92] J.-Y. Le Dimet, Enlacements d’intervalles et representation de Gassner, Comment. Math. Helv. 67(1992), no. 2, 306–315.

[Les98] C. Lescop, A sum formula for the Casson-Walker invariant, Invent. Math. 133 (1998), no. 3, 613–681.[Lev69a] J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110; addendum, ibid. 8 (1969), 355.

MR 0253348 (40 #6563)[Lev69b] , Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110; addendum, ibid. 8 (1969), 355.

MR 0253348[Lev94] J. P. Levine, Link invariants via the eta invariant, Comment. Math. Helv. 69 (1994), no. 1, 82–119.

MR 1259607[Lib82] A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J.

49 (1982), no. 4, 833–851. MR 683005 (84g:14030)[Lib83] , Alexander invariants of plane algebraic curves, Singularities, Part 2 (Arcata, Calif., 1981), Proc.

Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135–143. MR 713242[Lib86] , On the homotopy of the complement to plane algebraic curves., Journal fur die reine und ange-

wandte Mathematik 367 (1986), 103–114 (eng).[Lib89] , Invariants of plane algebraic curves via representations of the braid groups, Invent. Math. 95

(1989), no. 1, 25–30. MR 969412[Lib01] , Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory,

physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad.Publ., Dordrecht, 2001, pp. 215–254.

[Lib07] , Lectures on topology of complements and fundamental groups, Singularity theory, World Sci.Publ., Hackensack, NJ, 2007, pp. 71–137. MR 2342909 (2008i:14043)

[Lin01] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl.Ser.) 17 (2001), no. 3, 361–380. MR 1852950

[Lit79] R. A. Litherland, Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. SecondSussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 71–84.MR 547456 (80k:57012)

[Lit84] , Cobordism of satellite knots, Four-manifold theory (Durham, N.H., 1982), Contemp. Math.,vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 327–362. MR 780587 (86k:57003)

[Liv02] C. Livingston, Seifert forms and concordance, Geom. Topol. 6 (2002), 403–408 (electronic). MR 1928840

BIBLIOGRAPHY 89

[Liv05] , A survey of classical knot concordance, Handbook of knot theory, Elsevier B. V., Amsterdam,2005, pp. 319–347. MR 2179265

[Mac36] S. MacLane, Some Interpretations of Abstract Linear Dependence in Terms of Projective Geometry,Amer. J. Math. 58 (1936), no. 1, 236–240. MR 1507146

[Mag34] W. Magnus, Uber Automorphismen von Fundamentalgruppen berandeter Flachen, Math. Ann. 109 (1934),no. 1, 617–646. MR 1512913

[Mil62] J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147.[Mil66] , Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.[Moi81] B. G. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980),

Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 107–192. MR 644819[Mor93] S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993),

no. 3, 699–726.[Mur65] Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965),

387–422. MR 0171275[Neu87] W. D. Neumann, Splicing algebraic links, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8,

North-Holland, Amsterdam, 1987, pp. 349–361. MR 894301[Oka92] M. Oka, Symmetric plane curves with nodes and cusps, J. Math. Soc. Japan 44 (1992), no. 3, 375–414.

MR 1167373[Oka02] , A new Alexander-equivalent Zariski pair, Acta Math. Vietnam. 27 (2002), no. 3, 349–357,

Dedicated to the memory of Le Van Thiem (Hanoi, 1998). MR 1979814[Ore99] S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4,

779–810. MR 1679799[Ore02] , Classification of flexible M-curves of degree 8 up to isotopy, Geom. Funct. Anal. 12 (2002),

no. 4, 723–755. MR 1935547[OS80] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56

(1980), no. 2, 167–189. MR 558866[Pap57a] C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957),

1–26. MR 0090053[Pap57b] , On Dehn’s lemma and the asphericity of knots, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 169–172.

MR 0082671[Pap75] , Planar regular coverings of orientable closed surfaces, Knots, groups, and 3-manifolds (Papers

dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 261–292. Ann.of Math. Studies, No. 84.

[Per06] B. Perron, A homotopic intersection theory on surfaces: applications to mapping class group and braids,Enseign. Math. (2) 52 (2006), no. 1-2, 159–186.

[Pol10] M. Polyak, Alexander-Conway invariants of tangles, ArXiv e-prints (2010).[Rei39] K. Reidemeister, Durchschnitt und Schnitt von Homotopieketten, Monatsh. Math. Phys. 48 (1939), 226–

239.[Roh71] V. A. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Funkcional. Anal. i Prilozen.

5 (1971), no. 1, 48–60. MR 0298684 (45 #7733)[Ryb98] G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, Preprint

available at arXiv:math.AG/9805056, 1998.[Ryb11] , On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal.

Appl. 45 (2011), 137–148, Preprint available at arXiv:math.AG/9805056.[Sak95] M. Sakuma, Homology of abelian coverings of links and spatial graphs, Canad. J. Math. 47 (1995), no. 1,

201–224. MR 1319696 (96d:57008)[Sak08] Takuya Sakasai, The Magnus representation and higher-order Alexander invariants for homology cobor-

disms of surfaces, Algebr. Geom. Topol. 8 (2008), no. 2, 803–848.[Sak12a] T. Sakasai, A survey of Magnus representations for mapping class groups and homology cobordisms of

surfaces, Handbook of Teichmuller theory. Volume III, IRMA Lect. Math. Theor. Phys., vol. 17, Eur.Math. Soc., Zurich, 2012, pp. 531–594.

90 BIBLIOGRAPHY

[Sak12b] , A survey of Magnus representations for mapping class groups and homology cobordisms of sur-faces, Handbook of Teichmuller theory. Volume III, IRMA Lect. Math. Theor. Phys., vol. 17, Eur. Math.Soc., Zurich, 2012, pp. 531–594.

[Sat84] N. Sato, Cobordisms of semiboundary links, Topology Appl. 18 (1984), no. 2-3, 225–234. MR 769293

[Sei35] H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592. MR 1512955[Smo89] L. Smolinsky, A generalization of the Levine-Tristram link invariant, Trans. Amer. Math. Soc. 315

(1989), no. 1, 205–217. MR 931532 (89m:57021)[Suc11] A. I. Suciu, Fundamental groups, Alexander invariants, and cohomology jumping loci, Topology of al-

gebraic varieties and singularities, Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011,pp. 179–223. MR 2777821

[Suz03] M.i Suzuki, On the Magnus representation of the mapping class group, Proceedings of the Winter Work-shop of Topology/Workshop of Topology and Computer (Sendai, 2002/Nara, 2001), vol. 9, 2003, pp. 77–80.

[Suz05] M. Suzuki, Geometric interpretation of the Magnus representation of the mapping class group, Kobe J.Math. 22 (2005), no. 1-2, 39–47.

[SW00] D. S. Silver and S. G. Williams, Coloring link diagrams with a continuous palette, Topology 39 (2000),no. 6, 1225–1237. MR 1783855

[Tri69] A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264.MR 0248854

[Tro62] H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962),464–498. MR 0143201

[Tur78] V. G. Turaev, Intersections of loops in two-dimensional manifolds, Mat. Sb. 106(148) (1978), no. 4,566–588.

[Tur86] V. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97–147, 240.[Tur00] , Introduction to combinatorial torsions, Birkhuser Verlag, 2000.[Tur01] , Introduction to combinatorial torsions, Lectures in Mathematics ETH Zurich, Birkhauser Verlag,

Basel, 2001, Notes taken by Felix Schlenk.[Vir73] O. Viro, Branched coverings of manifolds with boundary, and invariants of links. I, Izv. Akad. Nauk

SSSR Ser. Mat. 37 (1973), 1241–1258. MR 0370605 (51 #6832)[Vir09] , Twisted acyclicity of a circle and signatures of a link, J. Knot Theory Ramifications 18 (2009),

no. 6, 729–755. MR 2542693 (2011b:57018)[VK33] E.R. Van Kampen, On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933), no. 1-4,

255–267. MR 1506962[Wad94] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), no. 2, 241–

256. MR 1273784[Wal99] C. T. C. Wall, Surgery on compact manifolds, second ed., Mathematical Surveys and Monographs, vol. 69,

American Mathematical Society, Providence, RI, 1999, Edited and with a foreword by A. A. Ranicki.MR 1687388

[Wir27] W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veranderlichen, Math. Ann.97 (1927), no. 1, 357–375. MR 1512366

[Zar29] O. Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a GivenBranch Curve, Amer. J. Math. 51 (1929), no. 2, 305–328. MR 1506719

[Zar31] , On the irregularity of cyclic multiple planes, Ann. of Math. (2) 32 (1931), no. 3, 485–511.MR 1503012

Documents

Intersection in low dimensional topologyIntersection in low dimensional topology Vincent Florens Habilitation a diriger des recherches Laboratoire LMAP UPPA { Universit e de Pau et