Flat Virtual Pure Tangles...virtual pure tangles modulo both kinds of R-II and R-III moves, as well as a braid-like variant in which only the braid-like R-II and R-III moves are allowed

Flat Virtual Pure Tangles

by

Karene Chu

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c⃝ 2012 by Karene Chu

Abstract

Flat Virtual Pure Tangles

Karene Chu

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2012

Virtual knot theory, introduced by Kauffman [Kau], is a generalization of classical knot

theory which interests us because its finite-type invariant theory is potentially a topo-

logical interpretation [BN1] of Etingof and Kazhdan’s theory of quantization of Lie bi-

algebras [EK]. Classical knots inject into virtual knots [Ku], and flat virtual knots [Ma1] [Ma2]

is the quotient of virtual knots which equates the real positive and negative crossings,

and in this sense is complementary to classical knot theory within virtual knot theory.

We classify flat virtual tangles with no closed components and give bases for its

“infinitesimal” algebras. As a corollary, we also obtain a classification of free virtual

tangles with no closed components. The classification of the former can be used as

an invariant on virtual pure tangles. In a subsequent paper, we will show that the

infinitesimal algebras are indeed the target spaces of any universal finite-type invariants

on the respective variants of flat virtual tangles.

ii

Contents

1 Introduction 1

2 Preliminaries 13

2.1 General Chord Diagram Algebras CD . . . . . . . . . . . . . . . . . . . 13

2.1.1 Graphs and the Gluing Operations . . . . . . . . . . . . . . . . . 13

2.1.2 General Chord Diagrams (CD) and its operations . . . . . . . . . 16

2.1.3 Another Way to Represent General Chord Diagrams . . . . . . . 18

2.1.4 On the Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.5 Subdiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.6 Subalgebras, Congruence Relations and Quotients of Free Algebras 22

2.1.7 Restriction of the Algebras to a Skeleton . . . . . . . . . . . . . . 23

2.2 Virtual Tangles, Flat Virtual Tangles,and Their Variants . . . . . . . . . 24

2.2.1 Subsets of Reidemeister Moves . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Descending Virtual Tangles . . . . . . . . . . . . . . . . . . . . . 29

2.3 The Associated Graded Spaces of Usual, Virtual, and Flat Virtual Tangles 30

2.3.1 Linear Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2 Finite-Type Invariant Theory . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Presentation of the Associated Graded Space of the free CA and free CD 34

2.3.4 The Associated Graded Space of general CA . . . . . . . . . . . . 36

2.4 The Associated Graded Space of vKG and fKG . . . . . . . . . . . . . . 37

iii

2.4.1 The Associated Graded Space of fPT . . . . . . . . . . . . . . . 39

3 Classification of Pure Descending Virtual Tangles 41

3.1 Generic Diagrams of Pure Descending Virtual Tangles . . . . . . . . . . . 41

3.2 The Sorting Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Sorting map is well-defined . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Classification of the Unframed Version: Adding Reidemeister I . . . . . . 54

3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Basis of Avf 58

4.1 Generalized Grobner Basis for Chord Diagram Algebra . . . . . . . . . . 58

4.2 A Partial Ordering on CD algebras . . . . . . . . . . . . . . . . . . . . . 65

4.3 Bases for Associated Arrow Diagram Algebras . . . . . . . . . . . . . . . 68

A Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5 81

Bibliography 85

iv

Chapter 1

Introduction

In this section, we first introduce two ways of presenting virtual tangles, define flat

virtual tangles and a variant of it, and state our first result, the classification for flat

virtual tangles in the case in which all .

We then define the associated infinitesimal algebras of the two variants of flat virtual

tangles, and state our second result: bases for each of them in the case of only open

labeled components.

While usual tangles are the planar algebra generated by the positive and negative

crossings modulo Reidemeister relations, virtual tangles are the generalization in which

the operations of the planar algebra are not necessarily planar anymore. More precisely,

a virtual tangle diagram is a finite set of crossings with the data of which crossing end is

connected to which other one but not how on the plane. In particular, the connections

among crossings may not be embeddable and necessarily intersect on the plane. These

intersections are called “virtual” crossings. For example, the left most diagram below is

a virtual tangle diagram where ends labeled with the same number are connected to one

another, and this virtual tangle diagram is drawn with the connections on the plane in

the middle below, in which the two crossings on the sides that are neither over nor under

1

Chapter 1. Introduction 2

are the virtual crossings. By definition, all ways of drawing the connections on the plane

are equivalent as virtual tangle diagrams.

4

3 7

6

1 8

6

2

3

2 8

7

5 4 5

_ + _

+

1

1

Figure 1.1: Example of a non-trivial virtual knot, the Kishino knot, drawn (L) as crossings

with labeled open ends which are glued together if they have the same label; (M) on the

plane where the non-circled crossings are virtual crossings; (R) as Gauss diagram where

the outer most circle is the skeleton of the virtual knot and the signed arrows represent

the crossings

Another natural way to present virtual tangle diagrams is by Gauss diagrams. These

are disjoint unions of lines and circles with signed arrows ending on them with a positive

arrow pointing from segment i to segment j representing a real “i over j” positive crossing

as shown below. The Gauss diagram corresponding to the example above is on the right

of Figure 1 above.

+ _

i j i i j j i j

Figure 1.2: The correspondence between real crossings and their Gauss chords.

Now, virtual tangles are the equivalence classes of virtual tangle diagrams modulo

finite sequences of Reidemeister moves, which are local relations as shown below in both

presentations. Usual tangles inject into virtual tangles.[ref]


Ɛ3

Ɛ2

Ɛ1 Ɛ3

Ɛ2

Ɛ1

R1 R2 R3

Ɛ

- Ɛ

Ɛ

R1 R2 R3

Figure 1.3: Reidemeister I, II and III moves. (L) on the plane; (R) Gauss diagram. ϵ’s

denote some specific signs.


To understand virtual tangles, one may first consider a quotient of virtual tangles by

the additional crossing-flip relation:

+i j i j i j i j

_= =

Figure 1.4: ”Flatness” relation for flat virtual tangles: a real crossing is equivalent to its

“flip.”

The special case of flat virtual tangles with only open labeled components can be

easily shown to be isomorphic to descending virtual tangles, which, as a set, is the subset

of virtual tangles with only descending crossings. A descending crossing is one in which

the “earlier” strand is over the “later” strand, where “earlier” and ”later” are w.r.t. the

orientation and/or numbering of the strands. Equivalently, in Gauss diagram terms, all

arrows point from earlier to later segments of the tangle.

Furthermore, since there are proper independent subsets among the set of all Reide-

meister moves, we can consider variants of virtual or even usual tangles for which not all

Reidemeister moves are imposed. For example, Reidemeister I relations can be dropped.

Also, following [?], we make a distinction, which descends to the quotient of flat virtual

tangles, between braidlike and cylic Reidemeister II (R-II) and III (R-III) moves:

R2b

R2c

R3b

R3c

Figure 1.5: Braidlike and cyclic Reidemeister II and III moves. The signs of crossings

can be forgotten since they do not matter in the definition.

Usual knots inject into braidlike virtual tangles.


In this paper, we consider a few variants of flat virtual tangles. We consider flat

virtual pure tangles modulo both kinds of R-II and R-III moves, as well as a braid-like

variant in which only the braid-like R-II and R-III moves are allowed. Then for both

variants, we consider the f ramed case in which R-I is not imposed, and the unframed

case in which R-I is imposed.

Our first main result is the classification of both the framed and unframed versions of

the variant of flat virtual pure tangles in which all R-II and R-III relation imposed. We

have not classified the braid-like variant. We present here the simpler one-component

case first and then the multi-component case.

Theorem 1.0.1 (Classification of Long Descending Virtual Knots, conjectured by Bar–

Natan). Long framed flat virtual knots Kvf are in bijection with the set of canonical

diagrams C1 whose general form is shown below in figure 1. A canonical diagram is

a diagram which does not contain bigons bounded by opposite signed crossings, as shown

inside the forbidden signs in figure 1, and that in the circuit algebra language, whose

skeleton strand has a point before which it is always over in any crossing in the planar

diagram language, or has only arrow-tails on it in the Gauss diagram language, and after

which it is always under, or has only arrow-heads on it.

Furthermore, C1 is in bijection with the set of all signed reduced permutations, where a

signed permutation is a set map ρ : 1, . . . , n −→ 1, . . . , n×+,− which projects

to the first components as a permutation and a reduced one satisfies the extra condition

that the image of pairs of consecutive numbers are not any of ((j,∓), (j + 1,±)), and

((j + 1,±), (j,∓)) for all j < n.

Long unframed flat virtual knots Kvf are in bijection with the subset of C1 which contain

no “R-I kinks,” as shown in the forbidden signs in figure 1.


...

...

PLANAR GAUSS

σ

... ...

where if R-1 imposed also: where

+/- +/-

-/+

-/+

if R-1 imposed also:

+/-

Figure 1.6: General form of the canonical diagrams of long descending virtual knots

drawn (L) as circuit algebra diagrams in which an unoriented strand means the strand

can be oriented in either direction, and the light gray dotted box bound a labeled region

in which the incoming strand on the right of the box stays and creates crossings in the

region until it exits on the left; (R) as Gauss Diagrams in which ϵ’s are signs of crossings,

a double arrow denotes a number of arrows and the box they start or end in denotes a

permutation of the arrows so that the incoming arrows are permuted by σ within the

box and emerge from the other side permuted.

Here is a sample diagram which is canonical for both a framed and unframed long

descending virtual knot. Another way to describe a reduced signed permutations is one

whose image of (1, . . . , n) does not contain opposite signed consecutive pairs.

- + + + +

-

σ(2) = (1 , + )

σ(3) = (2 , + )

σ(1) = (3 , - ) σ :

PLANAR GAUSS REDUCED SIGNED PERMUTATION

Figure 1.7: Example of the Canonical Form of a Flat Virtual Knot

Slightly more general is the following for the multi-component case:

Theorem 1.0.2 (Classification of Pure Descending Virtual Tangles). Pure framed (i.e.

R-I not imposed) descending virtual tangles of n components are in bijection with the


set of canonical diagrams Cn whose general form is shown below in figure, which are

characterized by the same two conditions as in the one component case but applied to all

n components.

Similar to the one component case, the unframed version of pure descending tangles is in

bijection with the subset of Cn which contain no “R-I kinks”.

...

...

...

...

...

...

...

1

2

n

1

2

n

n-1

:=...

... := Ɛ1 Ɛ2 Ɛk

1

...

...

...

...

2

n

PLANAR GAUSS

σ

where

where

+/- +/-

-/+ -/+

... Ɛ1 Ɛ2 Ɛk

permute by σ

1 2 k

... σ(1) σ(2)

+/-

NOTATION

σ(k)

if R-1 imposed also:if R-1 imposed also:

σ1,o σ1,i

σ2,o σ2,i

σn,o σn,i

Figure 1.8: Canonical diagrams of pure descending virtual tangles of n components drawn

(L) as circuit algebra diagrams; (R) as Gauss Diagrams. See caption above in figure 1

Here is an example of a diagram which is canonical for both framed and unframed

pure descending virtual tangles:

PLANAR GAUSS

1

2

3

1

2

3

+

+

+

+

+

+

-

-

Figure 1.9: Example of the Canonical Form of a Flat Virtual Pure Tangle

The above classification also descends to the free quotient:


Theorem 1.0.3 (Classification of Free Virtual Pure Tangles). Pure free virtual framed

tangles (i.e. R-I imposed) of n components are in bijection with the set of diagrams in

Cn but with all signs forgotten. In particular, the one component ones, i.e. free virtual

long knots, are in bijection with the set of reduced permutations.

For examples, take the Gauss diagrams in figures 1 and 1 but omit the signs of all

chords.

The proofs of all three theorems are similar and amount to showing that a sorting

maps exists and is well-defined. For examples of a sorting algorithm, see figure 2.

Now, our central interest in the finite-type invariant theory of an algebraic structure

is to find a universal, better yet also homomorphic, finite-type invariant from an algebraic

structure to the graded algebraic space associated to the filtration given by powers of the

augmentation ideal, which is the set of all formal differences of elements in A. Before we

ask this question, we want to understand the target space. The second part of this paper

gives a basis for the conjectured associated graded space of descending virtual tangles

with open and labeled components, and also of the braidlike variant.

In classical knot theory, the augmentation ideal is planar algebraically generated

by crossing flips. In virtual knot theory, the augmentation ideal can be generated by

differences between real crossing and no crossing, called the semi-virtual crossing. In

both the two languages:

+:= :=_ _ _

:=i j i j i j i j i j i j

+_

:=_ _

i j i j i j i j i j i j

Figure 1.10: Semi virtual crossings.


In analogy with chord diagram space modulo 4T relations for classical knots, the

associated graded space for braidlike virtual tangles is the space of arrow diagrams modulo

the 6T relation, the “infinitesimal Reidemeister III” relation. An arrow diagram is a

disjoint union of circles and lines with unsigned arrows ending on it, and the 6T relation

is as follows:

i j k i j k i j k

+ + _

i i j k j k i j k

:=where

Figure 1.11: The six-term (6T) relation. The square brackets denote commutators

The associated graded space for the virtual tangles including the cyclic Reidemeister

moves has the following extra relation called the XII relation:

_

i i j j

Figure 1.12: The XII relation

In particular, for long descending tangles, its augmentation ideal would be gener-

ated by descending semi-virtual crossings only, and so is spanned by descending arrow

diagrams Dvf .

1

2

3

Figure 1.13: (L) An arrow diagram on a skeleton consisting of a circle and a line; (M;R)

Descending arrow diagrams on n strands and on 1 strand

For braidlike framed long descending virtual tangles, the associated graded space Avfb

is descending arrow diagrams modulo the descending 6T relation; and for framed long


descending virtual tangles (including cyclic Reidemeister moves), the associated graded

space Avf is quotient of Avfb by the descending XII relation. The fact that Avfb and Avf

are the associated graded spaces will be proved in a second paper.


Theorem 1.0.4 (Basis of Avfb, associated graded space of the Long Framed Braidlike

Descending Tangles). A basis of Avfb(↑1↑2 . . . ↑n) is the set of descending arrow diagrams

in which all skeleton components first have only outgoing arrows and then have only

incoming arrows. Restricting to the ”long knot case,” Avfb(↑) has basis in bijection with

elements of the union of symmetric groups of all order∪

n Sn. Thus the dimension of

degree n subspace of Avfb is n! as computed up to finite order in [BHLR].

:=... ...

... 1

σ

2 k

σ(1) σ(2) σ(k)NOTATION

... :=

1

...

...

...

...

2

n

Figure 1.14: The form of a basis element of (L) Avfb(↑1↑2 . . . ↑n) and (R) Avfb(↑1). A

double arrow denotes a number of arrows and the box they start or end in denotes a

permutation of the arrows.

Theorem 1.0.5 (Basis for Avf ). A basis of Avf is the subset of the basis of Avf that

excludes diagrams with the following subdiagrams where the left skeleton segment precedes

the right one.

Figure 1.15: the illegal subdiagrams

Further directions are to analyze round flat virtual knots and to find a universal

finite-type invariant in the spirit of the Kontsevich Integral for flat virtual tangles that

respect some algebraic operations on it.


This paper is organized as follows. In section ??, we define general chord diagram

algebras and in terms of it the different variants of virtual and flat virtual knotted objects.

Then, section ?? gives the proofs of theorems 1.0.1 and 1.0.3, the classification of long

flat virtual knots and pure tangles; and section ?? gives the proofs of theorems 1.0.4

and 1.0.5, the bases of the infinitesimal algebras.

Acknowledgments

I am indebted to my advisor Dror Bar-Natan for the computational evidence [BHLR]

and conjecture on the main results of this paper, the idea of using the finger move in the

first proof, on top of his generosity with his time, resources, and care, and many hours of

inspiring discussion. I also learned the elegant concepts in the preliminary section from

him.

I am grateful for mathematical discussion with P. Lee, Z. Dancso, L. Leung, I. Ha-

lacheva, J. Archibald. In particular, P. Lee pointed out [BEER] which helped me under-

stand more about flat virtual braids, known as the “triangular group” in the paper. I

first learned about flat virtual knots in a lecture by V. Manturov in the Trieste summer

school on knot theory.

Chapter 2

Preliminaries

In this section, we first define the formal language of general chord diagram algebras CD

(sec 2.1)and then define virtual and flat-virtual knotted graphs in terms of it (sec 2.2).

The objects of this definition will be the well-known Gauss diagrams, but we also em-

phasize the algebraic structure among Gauss diagrams.

In the second half, we give a brief general introduction to universal finite-type invari-

ant theory of general chord diagram algebras as motivation to study of “infinitesimal”

algebras in theorems 1.0.5 and 1.0.4 (sec 2.3). We explain briefly the derivation of and

define the infinitesimal algebrasAv andAf of virtual and flat virtual pure tangles (sec 2.4)

as a general chord diagram algebra (sec 2.4). The algebraic structure of these general

chord diagram algebras are crucial in the proofs of theorems 1.0.5 and 1.0.4 (sec 4.1).

2.1 General Chord Diagram Algebras CD

2.1.1 Graphs and the Gluing Operations

Let us start with graphs. Graphs can be glued together to form larger graphs. Roughly

then, general chord diagrams have an underlying graph and the gluing operation on the

13

Chapter 2. Preliminaries 14

underlying graph will be its algebraic structure.

More precisely, we will loosen the definition of graphs to include also “loops” without

any vertices:

Definition 2.1.1. 1. What we call graph below is a finite (not-necessarily connected)

classical graph with no isolated vertex and possibly disjoint union with a finite num-

ber of loops with no vertices. We will consider these graphs with parametrization,

and call them parameterized graphs, where the parametrization is implemented

by ordering the set of all edges and orienting each edge.

2. For simplicity, an edge with univalent vertices on both ends is called a strand,

a “loop” without any vertex is still called a vertexless loop, and a multi-valent

vertex with the half-edges incident to it capped with univalent vertices is shortened

to vertex graph. We will also call the univalent vertices in a parameterized graph

an open end of the parameterized graph.

Clearly, on the set of parameterized graphs, there is a set of re-parametrization oper-

ations which involve re-ordering and re-orientating edges. (The ordering of edges will be

given by integer label from 1 to the total number of edges.)

Definition 2.1.2. Let Gi be parameterized graphs. Then, for N ≥ 1, an N -nary gluing

operation identifies pairs of oppositely oriented univalent vertices in the input graphs

G1, . . . , GN , according to some list L, then deletes the identified vertices (such that the

pairs of edges or pairs of half edges incident on them are respectively merged into single

edges or single loops with no vertices), and then parameterize the resulting graph. If L is

empty, this operation will simply return a re-parameterized disjoint union of the inputs.

Implementation–wise, any univalent vertex in a parameterized graph can be referred

to by the only edge e incident to it and whether this edge is incoming or outgoing, denoted


by a sign δ. Thus, the list L can be given by L = (i1, e1, δ1), (i1′ , e1′ , δ1′), . . . , (im, em, δm), (im′ , em′ , δm′)

where each triple (ik, ek, δk) denotes the (ek, δk) univalent vertex of the ithk input graph.

For the set of parameterized graphs that do not contain loops without vertices, we

can restrict to a subset of the gluing operations.

Definition 2.1.3. 1. Let G be a parameterized graph that does not contain loops

without vertices. The points on the edges of G can be ordered first by the edge-

ordering and then if two points are on the same edge by orientation. We call this

ordering the orientation of G.

2. An N -nary gluing-operation is orientation-preserving if the orientations of all

N input parameterized graphs remain unchanged in the output.

Remark 2.1.4. 1. Clearly, all parameterized graphs can be obtained by gluing opera-

tions on single strands and vertices.

2. The subsets of parameterized graphs obtained by gluing together only particular

kinds of vertices are closed under the gluing operations. For example, the subset

of parameterized graphs with no multi-valent vertices, i.e. disjoint unions of only

strands and loops.

3. Gluing a single strand onto any open end of a parameterized graph does not change

the graph except for possibly the parametrization.

4. We can consider vertices decorated with additional discrete data, for example,

vertices in which the incident edges have a cyclic ordering.

Here are some examples of simple binary graph-gluing operations:


, 1

1

2

1 ((1,1,-) , (2, 2, +)),

((2,2,-) , (2, 1, +)) 2

3 input 1 input 2 input 2

1 2

3

2 1

,

1

2 2

((2,1,-) , (1, 2, +)),

((1,2,-) , (2, 2, +)) input 2 input 1 input 2

1 2 2

1

2

2

3

1

input 1

1

((1,1,-) , (1, 1, +))

1

2

1 2

2

input 11

input 1

Figure 2.1: Some gluing operations on parameterized graphs.

2.1.2 General Chord Diagrams (CD) and its operations

Definition 2.1.5 (General Chord Diagram). A (general) chord diagram D is a pa-

rameterized graph S, (definition 1), called the skeleton of D, with finite pairs of marked

points on its edges connected by another type of edges called chords, considered up to

combinatorics, i.e. the marked points are considered only up to their ordering on an edge

and cannot move through any vertex on S. Both the vertices of the skeleton S and the

chords in D are allowed to have extra discrete data on them, for example, chords can be

directed and signed.

Definition 2.1.6 (General Chord Diagram Operations). Let DS be the set of all general

chord diagrams with skeleton S. An N -nary CD-operation is a map DS1×. . .×DSN −→

DS which performs an N -nary gluing operation (as in definition 2.1.2) on the skeletons

of the inputs, while doing nothing to the chords them. Accordingly, an orientation-

preserving CD-operation is one which performs an orientation-preserving gluing op-

eration to the skeletons.


Here is an example of CD-operation

.

1

input 1 input 2 input 2

1 2

3

2 1

2

input 2 input 1 input 2

1 2 2

2

3

1

input 1

1

1 2

2

input 11

input 1

: D D D1

2

3 1

2

, 1

1

2 2

3

1

2

3

1

2

3

: D D D1

2

1

2

1

2

,

1

2

1

2 2

1

: D D1

2

1

2

1

2 1

2

Proposition/Definition 2.1.7. 1. Given a set of general chord diagrams D, the

set of all general chord diagrams generated (resp. generated via orientation-

preserving operations) by D is the set of all outputs of the CD-operations (resp.

orientation-preserving CD-operations) with diagrams in D as inputs.

2. In particular, given a set v1, . . . , vm, χ1, . . . , χn where vi is a vertex graph and

χi is a chord diagram consisting of two strands and a decorated chord with one

end on each strand (see figure 2.1.3), called single-chord diagram, the set of

general chord diagrams generated by this set is exactly the set of all general chord

diagrams whose decorated vertices are all of the types in v1, . . . , vm, and whose

decorated chords are all of the types in χ1, . . . , χn. The set of general chord

diagrams generated via only the orientation-preserving CD-operations will contain


only chord diagrams on skeletons for which orientation is well-defined, i.e. skeleton

with no vertexless-loops.

Definition 2.1.8 (General Free Chord Diagram Algebra). 1. The

free chord diagram algebra CD⟨v1, . . . , vm, χ1, . . . , χn⟩ is the set of all chord dia-

grams generated via all CD-operations by the set of vertex graphs v1, . . . , vm and

the set of single-chord diagrams χ1, . . . , χn along with the CD operations on them.

2. Similarly, the free oriented chord diagram algebra−→CD⟨v1, . . . , vm, χ1, . . . , χn⟩

is the set of all chord diagrams generated via all orientation-preserving CD-operations

by the set of vertex graphs v1, . . . , vm and the set of single-chord diagrams

χ1, . . . , χn along with the orientation-preserving CD operations on them.

Remark 2.1.9. 1. The skeleton map S from CD-diagrams into parameterized graphs,

which forgets the chords of any CD-diagram and outputs its underlying skeleton,

commutes with all CD-operations and in this sense it is a forgetful “functor.”

2. The degree deg(D) ∈ Zn of a general chord diagram D is the ordered set of num-

bers of different types of decorated chords in D. This degree is additive under the

CD-operations.

2.1.3 Another Way to Represent General Chord Diagrams

There is another way to represent general chord diagrams simply by representing one

type of generators differently.

Namely, the single-chord diagram with chord decoration Ω can be replaced by a

tetravalent vertex graph where the vertex is decorated by Ω but also by an extra pairing


of the edges incident on it. Defining the skeleton of this tetravalent vertex to be the same

as that of the single-chord diagram it comes from, the CD-operations can then be defined

the same as before with the output considered up to the same level of combinatorics as

in the other representation, i.e. as graphs in this representation.

Ω

1

Ω

2 1 2

In the special case that the vertices are cyclically-ordered and all chords are decorated

by an extra binary bit, we can represent the generators as plane graphs: the vertex

graph generator can be presented by a vertex graph embedded on the the plane with

its edges ordered around the vertex on the plane according to the given cyclic ordering,

and the single-chord diagram by embedded tetravalent vertices with paired edges as

opposite edges on the plane in one of two way according to the binary bit decoration.

See figure 2.1.3.

ΩΩ

1 2 1 2

ΩΩ

1 2 2 1

1

2

3

1

2

3

1

2

3

1

2

3

2.1.4 On the Operations

First, let us define multiplication operators which are analogous to right- or left-multiplication

operators Rw : A → A in any associative algebra A:

Definition 2.1.10 (Multiplication operators). an N -nary operation in which N − M

inputs are already fixed can be seen as a M-nary multiplication operator which

“multiplies” the unfixed inputs by the fixed ones via the N -nary operator. We denote a

unary multiplication operator by θa where the subscript a indexes all information

so that θa = θa′ iff θa(D) = θa′(D) for all general chord diagrams D. The inputs and

outputs of θa are parameterized according to which algebra is in question.


Here is an example of multiplication operator θa : GS −→ GS′in a general chord

diagram algebra:

1

input input

2 1

2

input

2

2

3

1

input

1

: D D1

2

1

2

1

2

3

1

2

3

: D D1

2

1

2

1

2 2

1

Figure 2.2: A unary CD multiplication operator, denoted θa. Notice that θa is considered

up the sliding of the base skeleton ends on the output base skeleton to another vertex

(pass other solid segments) and the solid skeleton has ends that slide on the output

base skeleton up to vertices, and before touching another solid end or without making a

joint solid segment become nothing. i.e. without changing the information given for the

restriction

We make a few remarks on the CD-operations.

Identity/Trivial Operations Since gluing any strand with no chords onto any open

ends of a decorated chord diagram does not change the diagram, free general chord

diagram algebras have many different identity operations on them. Here is an

example of a CD identity/trivial operation:

Unary Operations The 1-nary CD-operations include all re-parametrization opera-

tion, while there is no non-trivial re-parametrization orientation-preserving CD-

operation.

Algebraic Structure on Operations The set of N -nary, N ≥ 1, CD-operations has


: D D

Figure 2.3: An identity CD-operator on the subset of decorated chord diagrams on a 1

strand skeleton

an algebraic structure. “Compatible” operations are composable since if an opera-

tion takes a certain subset of diagrams as input, then it can also take any operation

that outputs diagrams in that subset as an input. The set of N -nary operations is

closed under such composition by construction and is in fact generated via compo-

sition by the unary and binary operations alone.

Axioms Here are some axioms satisfied by the structure on the CD-operations. First,

any operation composed with a compatible identity operator is the same oper-

ation. Secondly, clearly an N -nary operation pre- or post-composed by a re-

parametrization operation is equal to another N -nary operation. Thirdly, any

N -nary operations can be a composition of an N -nary disjoint union operation

followed by a unary gluing operation. And most importantly, the composition also

satisfies the generalized associativity axiom that all ways of decomposing a trinary

operation into compositions of two binary operations are equal (and by induction

the same for N -nary operations).

2.1.5 Subdiagrams

Definition 2.1.11 (Subskeletons and Subdiagrams). A subskeleton S of a skeleton S

is one such that S is the output of a graph gluing operation (see definition 2.1.2) with S

as an input, or equivalently,

where restriction means

Similarly, a subdiagram d of a general chord diagram D is a general chord diagram


such that θa(d) = D for some unary multiplication operation θa (see figure ?? for examples

of θa). Equivalently, a subdiagram d of a diagram D is a restriction of D considered up

to re-parametrization of d.

G=

G=

G=

G

Figure 2.4: Equivalent ways of drawing the boundaries of the subdiagram G on a 1 strand

skeleton.

A subskeleton S of a skeleton S is one such that S is the output of a graph gluing

operation (see definition 2.1.2) with S as an input. Finally, we can define a partial

ordering on the set of all skeletons by s ≤ S if s is a subskeleton of S, and similarly on

the set of all diagrams in an algebra. Clearly, the skeleton of d is a subskeleton of the

skeleton of D, i.e. S(d) ≤ S(D) if d ≤ D.

2.1.6 Subalgebras, Congruence Relations and Quotients of Free

Algebras

Subalgebras of the free algebras are as usual subsets closed under all operations. The set

of all diagrams generated by some set of diagrams D1, . . . Dn, i.e. the set of all outputs

of all operations with only the diagrams Di as inputs is clearly a subalgebra.

We define a general algebra generated by and decorated crossings χ1, . . . , χn to be

the quotient of the free algebra modulo some congruence relations. Congruence rela-

tions are equivalence relations closed under all multiplication with other diagrams and

have at least the two major sources. First, a congruence relation can be generated by a

(generating) relations D = D′ where D,D′ are diagrams “of the same type,” D and D′

must have the same skeletons. Here, generation means applying all possible operations


simultaneously to both sides of the equation so that the set generated by D = D′ is

θa(D) = θa(D′) | θa a multiplication operator, or equivalently the set of all equations

relating diagrams which include D on the L.H.S. and D′ on the R.H.S. as subdiagrams

in the same way. Notice that any relation that involves “smoothings” of the crossings

(see figure 2.2.5) can be used to generate congruence relations in circuit algebras but

not in general chord diagram algebra. A circuit (resp. chord diagram-) algebra which is

a quotient by a finite set g1 . . . gm of congruence relations of this kind can be finitely

presented as CA⟨υ1, . . . , υm, χ1, . . . , χn | g1, . . . gk⟩ (resp. CD⟨χ1, . . . , χn | g1, . . . , gk⟩ on

specified sets of skeletons). Secondly, there are congruence relations which are proper

subsets of relations generated by an equation D = D′ which satisfy some extra condition

at the skeleton level, e.g. a specific open end of D (and correspondingly D′) has to be

eventually be glued to another specific open end even though there can be any crossings

or chords in between. For example, the double-delta move satisfied by the multi-variable

Alexander polynomial [NS1]. The extra condition amounts to a restriction of the set of

operations used for generation.

Remark 2.1.12. 1. In a CD-algebra, a relation relating diagrams of the same degrees

is called homogeneous, and if all generating relations are homogeneous, then clearly

the degree descends to the elements of the quotient.

2. There is a projection map on any circuit or general chord diagram algebra by

forgetting all decorations on crossings or chords.

2.1.7 Restriction of the Algebras to a Skeleton

A free algebra restricted to a skeleton S is the set G of all diagrams on any subskeleton

S of S along with the set of operations with inputs and outputs restricted to these


diagrams. For free and free orientation-preserving general chord diagram algebras, these

restrictions are easily described: the objects are G := ∪S≤SGS and the operations are

restricted to the proper subset of CD operations Op : GS1 × . . .GSN −→ GS where all

skeletons Si’s and S are subskeletons of S. For free circuit algebras, the restrictions are

less compatible with the parametrization of the operations. The inputs to any N -nary

CA-operation Op : Gm1,ϵ1× . . .GmN ,ϵN −→ Gm,ϵ need to be restricted to the eligible subset

of Gm1,ϵ1×. . .GmN ,ϵN such that the disjoint union of the skeletons of the N input diagrams

is a subskeleton of S, and then on any set of eligible inputs E ⊂ Gm1,ϵ1 × . . .GmN ,ϵN , only

a subset of all N -nary CA-operations will output a diagram in G.

Finally, we consider generation by local diagrams g ∈ G rather than by global dia-

grams in D, and by local relations g = g′ with g, g′ in G rather in D which also have the

same skeleton. We may also restrict to global diagrams D ⊂ G generated by g ∈ G. Tak-

ing a quotient by a set of congruence relations means imposing the congruence relations

in G and also in D by considering D as a subset of G.

Subalgebras, congruence relations, and quotients are defined analogously as before

but with the “restricted” operations just described.

2.2 Virtual Tangles, Flat Virtual Tangles,and Their

Variants

We can redefine virtual and flat virtual knot theories as general chord diagram algebras

in the last section:

Definition 2.2.1. Virtual knotted graphs vKG is the circuit algebra CA⟨χ+, χ−,Vertices |

“R-moves”⟩ with the same generators and relations, or with a more enriched algebraic

structure, it can be defined as the union∪

S a skeleton CD(S)⟨χ+, χ− | “R-moves”⟩ over

all skeleton graphs S of general chord diagram algebras generated by signed directed

chords χ+, χ− representing the crossings modulo Reidemeister relations. The subsets of


Reidemeister-relations will be discussed below in section ??.

To match convention that the positive crossing represents the projection of a right

handed crossing, we depict χ+ by the left incoming strand going over the right incoming

strand, and vice versa for χ−, and We will redraw the signed dotted chords as signed

directed chords as follow so that the chord points always from the over strand to the

under strand: The decorated chord diagram for vKG, or in the circuit algebra definition,

the CA-diagrams with crossings χ+, χ− drawn as signed directed chords on the skeleton,

are called Gauss diagrams.

+

1 21 2

+ -

1 21 2

-

1 2 1 2

Figure 2.5: Real positive and negative crossings.

Definition 2.2.2. Flat virtual knotted graphs fKG is the image of virtual Knotted

Graphs under the crossing decoration forgetful map defined in 2.1.5. which in this case

means the signs of the crossings are forgotten such that χ+ = χ− and is called the

”flatness relation, ” and is isomorphic to CA(↑1↑2 . . . ↑n)⟨χVertices | “flat R-moves”⟩

or⊔

S CD(S)(↑1↑2 . . . ↑n)⟨χ | “flat R-moves”⟩ where χ is the undecorated crossing (but

dotted chords)

Thus, flat virtual knotted graphs is the simplest circuit algebra with Reidemeister-

type relations. Note that the crossings that are generators in these algebras are what is

usually called the “real” crossings, and the virtual crossings are needed only when the

CA − diagrams with crossings represented by are drawn on the plane.

Definition 2.2.3. Usual/virtual/flat-virtual round knots/long knots/pure tangles are

the restrictions of the usual/virtual/flat-virtual knotted graphs to the respective skeletons

a circle, a line, and ↑1↑2↑n.


Our main subject in this paper will be two variants of flat virtual pure tangles (fPT )

and their associated graded spaces.

2.2.1 Subsets of Reidemeister Moves

Here are the circuit algebra relations, the Reidemeister-moves (R-moves), in the definition

of virtual knotted graphs. The “flat R-moves” are the same CA-diagrams but with the

over and under information forgotten, thus flat; or in the CD diagrams with the signs

omitted. We drew them in Gauss diagrams.i

Ɛ3

Ɛ2

Ɛ1 Ɛ3

Ɛ2

Ɛ1

R1 R2 R3

Ɛ

- Ɛ

Ɛ

R1 R2 R3

Figure 2.6: Reidemeister II and III moves, drawn modulo the labelling and orientation

of the skeletons and open ends. (L) circuit algebra;(M) decorated chord diagram where

the dot on the chord-ends mark the left incoming strands of the crossings; (R) Gauss

diagram where the arrow points from the over to under strand.

Here is the Reidemeister IV or vertex invariance relations just for completeness. We

will not need this.

VI

Figure 2.7: Vertex Invariance.

By variants of the different quotients of virtual knots, we mean the different algebras

in which different subsets of the Reidemeister moves are imposed.

Definition 2.2.4. The framed variant of virtual/flat virtual/free virtual knots is the

algebra generated by the respective diagrams modulo all Reidemeister moves but the R-I


moves.

The braid-like variant is an the quotient in which the cyclic R-II and R-III moves are

not imposed, where “cyclic” and “braid-like” are defined below in definition 2.2.6.

We now show that these variants are indeed different. First, Reidemeister I is in-

dependent of any R-II and R-III since it changes the number of crossings/chords of a

diagram by 1, but both R-II and R-III change the number of crossings/chords by an even

number. Secondly, the braid-like variants are indeed different from the quotients in which

all including the cyclic Reidemeister moves are imposed, since the cyclic R-II and cyclic

R-III moves cannot be realized as a sequence of only braid-like moves as shown below.

We need two definition first. Note that the definitions are independent of the signs and

directions of the crossings/chords and so descends to both the flat and free quotients.

Definition 2.2.5. The complete orientation-preserving smoothing map ϕ is a CA-map

from any circuit algebra CA⟨χ1, . . . , χn, v1, . . . vm | R⟩ to the circuit algebra CA⟨v1, . . . vm |

R⟩ generated only by vertices that replaces any decorated crossing χi by two strands that

switches the connection and relabel the skeleton:

Figure 2.8: An orientation preservation smoothing

Definition 2.2.6. A Reidemeister II or III relation generator g1 = g2 is cyclic if the

images ϕ(g1) and ϕ(g2) of the complete orientation-preserving map contain a close cycle

in its skeleton; otherwise, it is braidlike. A Reidemeister relation is cyclic (resp. braidlike)

if generated by a cyclic (resp. braidlike) generator.

Proposition 2.2.7. The set of all braid-like Reidemeister II and III relations is a proper

subset of all Reidemeister relations.


R2b

R2c

R3b

R3c

Figure 2.9: Cyclic and braidlike Reidemeister II and III moves up to labelling of strands

and open ends

Proof. Applying the complete smoothing map ϕ to both side of any braid-like R2- or R3-

move, we get that each incoming open end is connected to the same outgoing open end

on both sides. but this is not the case for both the cyclic R2- and R3- moves.

Thus, we can consider the braid-like usual/virtual/flat virtual knotted graphs uKGb/vKGb/fKGb

which has only braid-like Reidemeister relations.

Remark 2.2.8. 1. In the presence of braid-like R-moves, the cyclic R2 implies cyclic

R3.

R2c R3c R2c

Figure 2.10: R3c as a composition of braid-like Reidemeister moves.

2. We enumerate the braid-like and cyclic R2- and R3-moves up to shifts of the open

ends and cyclic shifts of the strand labels by counting quantities that are invariant

on both sides of the moves. For R2, there is a unique labelling of the strands which

can have three orientation combinations, two giving rise to cyclic R2 moves and

two braid-like. Thus, there are four flat R2-moves. Then for each of these, there

are two choices of picking the top strand to form the (non-flat) R2-moves.

For R3, there are 2 cyclic orderings of the three skeleton strands each of which has

4 different orientation combinations, One of which results in a cyclic R3 move and


three in braid-like moves. The braid-like moves are distinguished by which of the 3

vertices of the triangle has one strand coming in and one coming out. Thus, there

are eight different flat R3-moves. For each of these flat R3 moves, there are 3× 2

to associate top, middle, bottom to each strand to form the R-moves.

In this paper, we classify flat virtual pure tangles with both braid-like and cyclic

moves

2.2.2 Descending Virtual Tangles

Definition 2.2.9. Given any skeleton S with no closed loops, the descending virtual

knotted graph on S is the free orientation-preserving CD-algebra on S generated by

descending crossings, crossings in which the “earlier” segment is always over the ”later”

one modulo the descending version of the relations for virtual knotted graphs. “Earlier”

and “later” are with respect to the ordering in definition 1.

+

1 21 2

-

1 21 2

Figure 2.11: Descending crossings. Both arrows point from strand 1 to 2.

Proposition 2.2.10. The forgetful projection π : vPT → fPT has a right (inverse l

which maps the generator, a flat crossing, to a descending crossing. This map respects

all the CD operations which does not change the order of the labels of the input and

the output, so the CD-subalgebra of descending virtual pure tangles is or general chord

diagram algebraically isomorphic to the CD of flat virtual pure tangles with a restricted set

of operations, namely the embedding and relabelling operations that preserve the ordering

of the input strands.

Proof. The map l is well-defined since it sends any Reidemeister relation in fPT to

one in vPT . (Notice this is not true if d maps any flat crossing to a positive crossing).


Clearly, π l = Id, and l is surjective onto the subset of descending virtual pure tangle

diagrams.

2.3 The Associated Graded Spaces of Usual, Virtual,

and Flat Virtual Tangles

2.3.1 Linear Extension

Much like a semigroup can be extended linearly to an associative algebra, any circuit

algebras or enriched chord diagrammatic algebras CA(S)⟨g1, . . . , gn | r1, . . . , rn⟩ described

above can be extended freely linearly over any field K.

The set of objects of the extended algebra is the K-vector space spanned by the

objects, i.e. equivalence classes of CA diagrams, of the CA.

The set of operations becomes the K-vector space spanned by the original set of

operations extended to N -linear maps on tensor products KG(S1) ⊗ . . . ⊗ KG(SN) of

vector spaces of diagrams G(S) parametrized by the skeleton S, and consequently, the

set of multiplication operators is the K-vector space spanned by linearly extended CA-

multiplication operators θa. Note that linear combinations of diagrams on different

skeletons may not be input into any operation.

A set of objects in the linearly-extended algebra now generates an ideal via the

linearly-extended multiplication operators.

In particular, any congruence relation generated by the equation g = g′ with g, g′ ∈

G(S) is lifted to the ideal generated by g − g′. As usual, the quotient of the extended

“algebra” by an ideal retains the algebraic operations from the free “algebra.”

An equivalent way to extend a circuit algebra or an general chord diagram algebra is


to extend the free algebras linearly as above and then quotient out by the ideals gener-

ated by gi − g′i where g = g′ generates a congruence relation in the original algebras.

Classifying the non-extended algebras is equivalent to finding a vector space basis for the

linearly-extended algebras.

In the following, we will abuse notation and denote the linearly-extended circuit and

enriched chord diagrammatic algebras by the same names CA and CD, and the special

cases of virtual and flat-virtual pure tangles by vPT and fPT respectively, as for the

non-extended versions.

2.3.2 Finite-Type Invariant Theory

In this section, we introduce the theory of finite-type invariants on circuit and enriched

chord-diagram algebras following Bar-Natan. This motivates our study of so-called in-

finitesimal algebras, in particular those associated to vPT and fPT –Avb, Av, and Avfb,

Avf respectively, as the target space of any “universal finite type invariant.”

Note that the following definitions generalize those in the case of usual associative

algebras, but also can be generalized to much more general algebraic structures.

Some standard definitions for a filtered vector space V = I0 ⊇ I1 . . ..

Proposition/Definition 2.3.1. 1. The completed associated graded vector space as-

sociated to V is

Gr V := I0/I1 ⊕ I1/I2 ⊕ . . . .

A homogeneous element in Gr V is one that belongs to only 1 direct summand

In/In+1, and the degree of such an element is n.

2. There exist non-canonical linear maps Z : V → Gr V such that Gr Z : Gr V → Gr V

is the identity map. If I∞ :=∩

n In is non-zero, then Gr Z does not depend on


Z |I∞ . If V/In is finite-dimensional, then it is isomorphic to I0/I1 ⊕ . . . In−1/In.

If V is infinite-dimensional, then Z is in general neither surjective nor injective.

Proof. We construct a map Z. Choose a sequence of linear section maps γi : Ii/Ii+1 →

Ii | i ∈ N. Notice there is no canonical choice for this. Let πi : V → V/Ii be the

projections. Define

Z = π1 ⊕ π2 (Id− γ0 π1)⊕ π3 (Id− γ1 π2 (Id− γ0 π1))⊕ . . . .

Then Z |Ii descends to the identity map on Ii/Ii+1. The rest are straightforward checks.

Let T be a CA or CD(S) and KT be its linearly-extension and “operations” be CA-

or CD-operations accordingly.

Proposition/Definition 2.3.2. 1. Let I be an ideal in KT . For n > 1, define the

nth power of I to be the vector space spanned by all outputs of operations with at

least n inputs in I, and denote it In. Then In is an ideal in KT , and in particular

in In−1.

2. With respect to the filtration of KT by the powers of I: KT ⊇ I ⊇ I2 ⊇ . . .,

all CA or CD operations are filtered, i.e. the output of an operation with inputs

respectively in Ik1 , . . . , IkN belongs to Ik1+...+kN .

3. Denote the associated graded space w.r.t. to the KT with the filtration by powers

of an ideal I by Gr IKT . Any N -nary CA or CD operation on T induces an N -nary

operation on Gr IKT . These are graded, i.e. the output of any induced operation

with homogeneous inputs of degree k1, . . . kN is of degree∑N

i=1 ki, and satisfy the

same axioms, such as generalized associativity, as the CA or CD operations that

induce them.


Proof. 1. Any multiplication operator θa on any D ∈ KT which is the output of an

operation with N inputs in I can be written as the output of an new operation

with the same N inputs in I.

2. By the definition of the powers of the ideals.

3. Any operation induced by a CA or CD operation will output an element in Ik1+...+kN/Ik1+...+kN+1

when the inputs are from Ik1/Ik1+1, . . . , IkNIkN+1 respectively. The induced op-

eration is well-defined since the CA operation with an ith input in a higher power

of I than Iki will output an element in a power of I higher than Ik1+...+kN . That

these induced operations satisfy all the axioms of the CA operations follows from

that the linearly-extended CA operations satisfy the same axioms with inputs of

linear combinations of CA elements satisfy the axioms with inputs in KT , not just

a linear extended version of the axioms.

The theory of finite-type invariants is the study of maps between the filtered and

the associated graded spaces w.r.t to the filtration by a specific canonical ideal, the

augmentation ideal.

Proposition/Definition 2.3.3. 1. In any CA or CD, there exists a canonical ideal,

called the augmentation ideal, which is the vector space ⟨D−D′ | D,D′ ∈ G(S)for any skeletonS⟩

spanned by formal differences of objects on the same skeleton in K, (i.e. equivalence

classes of diagrams on the same skeleton with the same ordering of external legs if

T is a CA, and on the same enriched skeleton if T is a CD. )

2. Let I be the augmentation ideal. We call a linear map Z : KT → Gr IKT as

in proposition 2 an expansion, also known as a universal finite-type invariant. A

homomorphic expansion is an expansion that respects all operations on KT .

Proof. The augmentation ideal is indeed an ideal since any multiplication operator θa


on the difference D −D′ of two objects of the same kind gives θa(D) − θa(D′), again a

difference of two objects of the same kind.

2.3.3 Presentation of the Associated Graded Space of the free

CA and free CD

We now find a presentation of the associated graded space of any free circuit-algebra

KCA⟨v1, . . . , vm, χ1, . . . , χn⟩ or free enriched chord-diagram algebraKCD(S)⟨χ1, . . . , χn⟩.

In the following we use FT for either algebra and “diagrams” and ’“operators” for either

CA- or CD-diagrams and operators according to context.

First, the generators of the augmentation ideal.

Proposition 2.3.4. The augmentation ideal IF of any linearly-extended free circuit or

enriched chord-diagram algebra, KCA⟨v1, . . . , vm, χ1, . . . , χn⟩ or KCD(S)⟨χ1, . . . , χn⟩,

is generated via the multiplication operators θa in the respective algebra, by the following

vectors χi := χi−S(χi), called the semi-virtual crossings/chords, one correspond to each

type χi of decorated crossings/chords:

+:= :=_ _ _


+_

:=_ _


Figure 2.12: Semi-virtual crossing. (L) as planar diagram; (R) as enriched chord diagram.

The

Proof. In both types of algebra, any difference of diagrams of the same kind can be

written as a telescopic summation of differences of diagrams which “differ by only one

crossing or chord”, i.e. θa(χ) − θa(S(χ)) where θa is a multiplication operator, χ a

generator and S(χ) is the skeleton of χ. So a spanning set of the augmentation ideal is

the set of all multiples of the semi-virtual crossings.


Proposition 2.3.5. The set of all diagrams with only semi-virtual crossings/chords χi

forms a basis for FT .

Proof. These diagrams clearly span since any diagrams in the free circuit algebra can

be written as linear combinations of diagrams with only semi-virtual crossings using the

inverse formula χi 7→ S(χi) + χi. To show linear independence, order original basis first

by the number of chords in it and then by a random ordering among the finite number of

diagrams with the same number of crossings, and observe that each element of the new

basis when written relative to the original basis using χi 7→ −S(χi) + χi has exactly 1

leading term which is simply the same diagram with all semi-virtual chords χi replaced

by the corresponding chords χi, and these leading terms are all different.

Relative to the new basis consisting of diagrams with only semi-virtual chords/crossing,

Proposition 2.3.6. 1. For all n ≥ 0, the nth power of the augmentation ideal InF has

basis the set of all diagrams with at least n semi-virtual crossings/chords. Thus,

the quotient InF/In+1

F for all n has basis the set of all equivalence classes with

representatives being diagrams with exactly n semi-virtual crossings/chords.

2. The graded space GrIFFT associated to the filtration of the free circuit or enriched

chord-diagram algebra by powers of the augmentation ideal IF is again the free CA-

or CD(′cS)- algebra KCA⟨v1, . . . , vm, χ1, . . . , χn⟩ or KCD(S)⟨χ1, . . . , χn⟩ gener-

ated by the semi-virtual crossings or chords.

3. Z : FT → Gr IFFT defined by the change of basis Z(χi) = cS(χi) + χi is a

homomorphic expansion.

Proof. These are all simple checks.


Remark 2.3.7. This is a direct generalization of the case of a free finitely-generated monoid

⟨x1, . . . , xn⟩. The augmentation ideal IF is generated by the differences xi := xi − 1, the

associated graded space w.r.t the filtration by powers of IF is the free finitely generated

monoid ⟨x1, . . . , xn⟩, and Z(xi) = 1 + xi is a homomorphic expansion.

2.3.4 The Associated Graded Space of general CA

We now turn to the question of determining the associate graded space of quotients of

free circuit- or enriched chord-diagram algebras, KCA⟨v1, . . . , vm, χ1, . . . , χn | r1, . . . , rk⟩

or KCD(S)⟨χ1, . . . , χn⟩ | r1, . . . , rk⟩, but we restrict to quotients by relations g = g′

between two diagrams of the same kind. We will denote such an algebra with T .

The powers of the augmentation ideal in T is by definition In := (InF +R)/R where

IF is the augmentation ideal of the free algebra, and R is the ideal generated by the

relations r1, . . . , rk. Then by many isomorphism theorems, each summand In/In+1 of

the associated graded space is:

In/In+1 = ((InF +R)/R)/((In+1

F +R)/R)) = (InF +R)/(In+1

F +R)

= InF/((In+1

F +R) ∩ InF ) = (In

F/In+1F )/((R∩ In

F + In+1F )/In+1

F )

We know from above that InF/In+1

F has basis all diagrams with exactly n semi-virtual

crossings/chords, so our main task in finding GrT is to find Rn := (R∩InF +In+1

F )/In+1F

for all n.

Proposition 2.3.8. 1. The subspace R := ⊔nRn is an ideal in the CA- or CD(S)-

algebra GrFT .

2. GrT = GrFT /R as a CA algebra.

3. For each defining generating relation ri of T , if under the projection FT → FT ⊕

FT /IF ⊕ FT /I2F ⊕ . . . has the first non-zero term in FT /In

F , then ri + InF is a


generating relation of GrT . In general, these may not generate all of R.

Proof. 1. R is closed under the induced operations in GrFT since the CA-product of

a relation r in InF with any element v in Im

F results in relation in In+mF .

2. The induced operations on GrT coincides with the induced operations on GrFT /R

3. From definitions.

2.4 The Associated Graded Space of vKG and fKG

We want to determine the associated graded space of our specific examples of the usual

and braid-like versions of virtual knotted graphs vKG and flat virtual knotted graphs

fKG, as well as of their restrictions to the pure virtual/flat-virtual tangles.

First, recall vKG is the quotient of the free circuit algebras generated by vertices and

two types of crossings χ+ and χ− by the Reidemeister relations and fKG is a quotient

of vKG by the flatness relations. This implies that Gr vKG and Gr fKG are quotients of

the free circuit algebra generated by the two respective semi-virtual crossings (on top of

the vertices):

+:= :=_ _ _


+_

:=_ _


Figure 2.13: Positive and negative semi-virtual crossings χ+ and χ−

Following proposition 3 and as in [GPV], we project the Reidemeister II, III and

flatness relations to the lowest degree by writing the crossings in terms of the semi-

virtual crossings, χ± 7→ S(χ±) + χ± and obtain the following generating relations for

GrvKG.


The Reidemeister II moves, both braid-like and cyclic, give between the two generators

of GrvKG so that we can eliminate the negative semi-virtual crossings chi from now on

in the presentation.

+=

--

Figure 2.14: Relation between + and − arrows in the same direction.

The Reidemeister III moves, with the four different orientations and the two cyclic

orderings of the three strands, gives the same degree two 6T relation (in terms of only

positive semi-virtual chords) in figure 1.

Also, the Reidemeister IV relation gives the vertex invariance relation:

= 0V.I.:

For fKG, the extra flatness relation gives the following “infinitesimal” flatness relation

between the two possible positive semi-virtual crossings on the two-strand skeleton:

+=

-FLATNESS:

Now, there are relations in GrvKG and GrfKG not generated by the “lowest or-

der terms” of a generating relation. From the difference of a braid-like- and a cyclic-

Reidemeister II move:

we obtain the XII relation in figure 1 for the versions of GrvKG and GrfKG with also

the cyclic Reidemeister moves


Thus, the following circuit algebras map into the respective associated spaces:

Avb := CA⟨v1, . . . , vn, χ | 6T, V I⟩ −→ GrvKGb

Av := CA⟨v1, . . . , vn, χ | 6T,XII, V I⟩ −→ GrvKG

Avfb := CA⟨v1, . . . , vn, χ | 6T, V I, F latness⟩ −→ GrfKGb

Avf := CA⟨v1, . . . , vn, χ | 6T,XII, V I, F latness⟩ −→ GrfKG

2.4.1 The Associated Graded Space of fPT

Now, as before, when we restrict to the case of pure tangles, there is a splitting map:

Proposition 2.4.1. The projection from Av(b) onto Avf(b) has a right inverse given by

the section map mapping each equivalence class of a flat crossing to the representative

diagrams containing only descending chord, one that points from an earlier to a later

segment in the skeleton. Thus as vector spaces, the subalgebra of descending arrow dia-

grams is isomorphic to the CD algebra of flat arrow diagrams Avf(b). The section map

commutes with all operations that preserves descendingness.

Proof. The map is well-defined since any 6T and XII relation gets mapped to a descending

relation.

1 2 n

...

3

+ + 6T:

1 2 n

...

3

+ + _ 6T: XII:,

Avfb

: =

Avf

: =

Figure 2.15: Sumary of definitions of Avfb(↑1 . . . ↑n) and Avf(↑1 . . . ↑n). The strands are

in descending order from left to right in the 6T relations.


In a subsequent paper we will show that the above are defining relations for the

associated graded space of flat virtual pure tangles, Avfb(↑1 . . . ↑n) ∼= GrfPT b and

Avf(↑1 . . . ↑n) ∼= GrfPT . In section 4.3, we give bases for these by giving bases for the

descending arrow diagrams.

.

Chapter 3

Classification of Pure Descending

Virtual Tangles

Having established that pure flat virtual tangles are equivalent to pure descending virtual

tangles in Section 2.2.2, we present in this section the classification of pure descending

virtual tangles and its proof. Recall from Section 2.2.2 that pure descending virtual n-

tangles is the subset of virtual tangles with only descending crossings on n open oriented

strands labeled from 1 to n. In this section, we will use “knots” and “pure n-tangles” to

stand for long descending virtual knots and pure descending virtual n-tangles respectively,

and “diagrams” to mean both the CA-diagrams and the Gauss diagrams representing pure

descending virtual tangles.

3.1 Generic Diagrams of Pure Descending Virtual

Tangles

In this subsection, we describe the general form of pure descending virtual tangle dia-

grams. First, a few definitions to describe the diagrams:

Definition 3.1.1. An interval of the skeleton of a pure descending virtual tangle is called

41

Chapter 3. Classification of Pure Descending Virtual Tangles 42

an over (resp. under) interval if all of its subintervals that take part in crossings are

the over strands in the crossings. A maximal over (resp. maximal under) interval is

an over (resp. under) interval preceded and followed immediately by an under (resp.

over) interval or by the beginning or end of the strand. An illegal interval is an interval

consisting of first a maximal under interval and then a maximal over interval. These are

illustrated below in Figure 3.1.

For clarity, we adopt the following conventions in all diagrams in this paper: we will

color the over interval of a crossing black and the under interval grey; in a CA diagram,

an interval not explicitly oriented means it can be oriented either ways; and in a Gauss

diagram, an unsigned Gauss arrow means it can have either sign. Also, in both the CA-

and Gauss diagrams, we use the ’“thick band” notations to represent multiple strands or

arrows as below:

...

:= ... Ɛ'

Ɛ

GAUSS

Ɛ1 Ɛ2 Ɛk = : Ɛ Ɛ' ...

Ɛ1' Ɛ2' Ɛm'

CA

...

Figure 3.1: An illegal interval, the skeleton interval within the square brackets, in (L)

circuit-algebraic- and (R) Gauss diagram languages. Within the illegal is first a maximal

under interval (in light gray) followed by a maximal over interval (in black). Any subin-

tervals of the maximal under (resp. over) is an under (resp. over) interval. The interval

preceding (resp. following) this illegal interval is either an over (resp. under) interval or

the beginning (resp. end) of the skeleton strand. Shown in the Gauss diagram language

is the case in which the illegal interval is between an over and an under interval. In the

Gauss diagrams, the half arrows have their other ends on other parts of the skeleton.

With this coloring, a pure descending virtual knot diagram has the following form:


...

...

...

...

...

...

...

1

2

n

n-1 :=

...

1

σ

2 k

σ(1) σ(2) σ(k)NOTATION

...

...

... ... ...

Ɛ1 Ɛ2 Ɛk

CA GAUSS

Figure 3.2: A generic diagram for a pure descending virtual knot. In the CA-diagram

(L), the light gray dotted boxes bound labeled regions in which the incoming strands on

the right of the box stays and creates crossings in the region until it exits on the left. In

the Gauss diagram (R), a double arrow denotes a number of arrows and the box they

start or end in denotes a permutation of the arrows so that the incoming arrows are

permuted by σ within the box and emerge from the other side permuted.

Note that any long descending virtual knot with at least one crossing has a skeleton

that starts with a maximal over interval and ends with a maximal under interval and

alternates between maximal over and under intervals in between. Also, due to descending-

ness, the ith maximal under-interval can only be under the first ith maximal over-intervals,

forming the crossings only within the region labeled “i” in the CA-diagram. The number

of illegal intervals corresponds to the number of boxed regions minus one. For a pure

n-tangle, a generic diagram is simply a long knot diagram with n−1 cuts on the skeleton

and with the resulting n components labelled “1” to “n” in order.

Remark 3.1.2. There are two parameters on the set of pure descending virtual tangle

diagrams: the number of illegal intervals, N (D), and the number of crossings, χ(D)

(whereD is a pure tangle diagram). Both are non-negative for all diagrams. Furthermore,

the number of crossings is bounded below by χ(D) ≥ N(D) + 1, since in the Gauss


diagram language, each of N (D) illegal intervals in a diagram D must have at least one

arrow-head and one arrow-tail, summing to 2N half arrows within the illegal interval, and

the beginning of the first strand and the end of the last strand must have one arrow-tail

and one arrow-head respectively. And this bound is attained by the following diagram:

Ɛ1 Ɛ2 ƐN+1 Ɛ3 ƐN

3.2 The Sorting Map

We start presenting the proof of theorems 1.0.1 and 1.0.3. Refer to page 5 for the state-

ment of the theorems and the definitions of canonical diagrams, forbidden subdiagrams

and reduced signed permutations.

We first show the bijection (in theorem 1.0.1) between the canonical diagrams C1 for

long descending virtual knots and reduced signed permutations, and then describe a sort-

ing map S that chooses a canonical representative diagram for each class of equivalent

pure descending virtual tangle diagrams.

Proposition 3.2.1. The set of one-component canonical diagrams C1 is in bijection with

the set of reduced signed permutations.

Proof. Consider a canonical diagram C with n arrows in the Gauss Diagram language.

Label the arrow-tails by 1, 2, . . . , n in increasing order from the start of the knot, and label

the arrow-heads similarly beginning with the first arrow head. Then construct a reduced

signed permutation ρ from the diagram by ρ(i) = (j, ϵ) where j and ϵ are respectively the

arrow-head label and the sign of the arrow with tail labeled i. There being no available

R2-sorts, or equivalently no subdiagrams in the forbidden signs in figure 1, translate to

the restriction that the image under ρ of pairs of consecutive numbers are not any of


((j,∓), (j + 1,±)), and ((j + 1,±), (j,∓)) for any j < n . The inverse of this map is

obvious.

First, we introduce the finger move, F-move:

-Ɛ

-Ɛ Ɛ

Ɛ

δ δ'

δ δ' δ δ'

δ δ'OR

CA GAUSS

F

Figure 3.3: Finger move. In the Gauss diagram language, there are two resulting diagrams

depending on the relative orientations of the two vertical strands in the CA diagram. δ’s

and ϵ’s are signs.

Proposition 3.2.2. The set of all F -moves and R2-moves is equivalent to the set of all

R3-moves and R2-moves.

Proof. R3-moves are generated by R2-moves and F -moves, as shown in the following

figure. Similarly, F -moves are generated by R2- and R3-moves.

F

R2R3

Figure 3.4: (L) an R3-move generated by an R2- and an F-move. Half of the R3-moves

are represented by this diagram, the other half are represented by the up-down-mirror

image of this diagram; (R) an F-move is generated by an R3- and an R2-moves.

Corollary 3.2.3. To show that a map on T Dvf descends to a map on T vf, it suffices to

show that the map is well defined under the finger moves and R-2 moves.


From now on we only use the planar looking CA-diagrams because the Gauss diagrams

have become too complicated and they can be constructed easily.

And now define two local sorting moves which will be used in putting a generic

diagram into its canonical form.

Definition 3.2.4. The sorting group-finger-move, GF-sort, and the sorting R2-move, R2-

sort, are the following single-direction moves that take place inside the squared region,

called the sorting site:

GF R2

Figure 3.5: (L) GF-sort; (R) R2-sort. An over (resp. under) thick band denotes a number

of over (resp. under) strands as in figure 3.1.

Remark 3.2.5. 1. GF-sort is generated by single sorting F-moves and so is generated

by R2 and R3-moves.

2. GF-sort switches the order of the maximal over interval and maximal under intervals

within the illegal interval, thus decreasing the number of total illegal intervals by

1, even if lengthening the illegal intervals that precedes or follows the one at the

sorting site.

3. GF-sort increases the number of total crossings by 2n > 0 of the diagram.

4. R2-sort decreases the number of total crossings by 2, and either does not change

or decreases the number of total illegal intervals by at most 2.

Some more terminology for the definition of the sorting map.

Definition 3.2.6.


1. A sorting move is available in a diagram D if a subdiagram of D is equal to the

L.H.S. of the sorting move. This subdiagram is called the sorting site in D for the

sorting move;

2. Two sorting moves s, t overlap if in the intersection of their sorting sites, there is

at least a crossing.

3. A sort sequence S on a diagramD is a finite sequence of sorting moves sk. . .s2s1

such that for each i, si is an available move on the diagram si−1 . . . s2 s1(D).

4. A terminating sort sequence on D is a sort sequence T such that T (D) has no

available sorting moves.

We can now characterize the set of canonical diagrams C to be all pure descending

virtual tangle diagrams with zero illegal intervals and no R2-sorting sites.

Definition 3.2.7. Define the sorting map on the set of all pure descending virtual tangle

diagrams T Dvf to be

S : T Dvf −→ T Dvf

D 7−→ sk . . . s2 s1(D)

where sk . . . s2 s1 is any terminating sort sequence on D.

Example 3.2.8.

See section?? for examples.

Proposition 3.2.9.

1. S is generated by Reidemeister-moves;

2. S is defined, i.e. the algorithm terminates


3. For any pure tangle diagram D, S(D) ∈ C ∈ T Dvf

Proof. 1. Both GF- and R2 sorts are a finite sequence of Reidemeister-moves;

2. only finite number of GF-sorts can be performed since a GF-sort decreases the

parameter ND (the number of illegal intervals) by 1 and R2-sorts do not increase

ND. Since the number of GF-sorts are finite, at the point in any sorting algorithm

when all GF-sorts are performed, only finite R2-sorts can be performed since it

decreases the parameter χD by 2;

3. the result of any terminating sort sequence has no illegal intervals and no R2-sorting

sites.

Lemma 3.2.10. S : T Dvf −→ T Dvf descends to a bijection S : T vf −→ T vf between pure

descending virtual tangles and the set of canonical diagrams Cvf defined in theorems 1.0.1

and 1.0.3 on page 5.

Proof. We need to show that S is well-defined under choices of terminating sorting

sequences, and well-defined under Reidemeister-moves, and is bijective into the set of

canonical diagrams C. Well-definedness of S follows from lemmas 3.3.1 and 3.3.2 in the

next section. It remains to show bijectivity, but surjectivity follows from the fact that a

canonical diagram does represent a pure descending virtual tangle and injectivity follows

from the fact that S applied to any canonical diagram results in the same canonical

diagram.

3.3 Sorting map is well-defined

This section is the main part of the proof of lemma 3.2.10, divided into lemmas 3.3.1

and 3.3.2


Lemma 3.3.1. S is well-defined under choices of different terminating sort sequences.

Proof. We proceed by a two-dimensional induction on (N (D), χ(D)), the number of ille-

gal intervals and the number of crossings of a diagram D ∈ T Dvf.We will first show that

S(D) is well-defined for all the diagrams D in the “column” N = 0 using an induction on

the variable χ, and then assuming the induction hypothesis for all “columns” N (D) ≤ n,

show the statement for the “column” N = n+1 by another an induction on χ. In all in-

duction steps below, we will use one of the two following general arguments.We will call a

region in the inductive domain where the statement is already true, either by hypothesis

or by proof, a truth region. First, for the case when a diagram D has only 1 available

sorting move, s, the sorting move s on D will result in a diagram in a truth region, i.e.

any terminating sorting sequence on s(D) gives the same resulting diagram. This then

implies that any terminating sorting sequence on D itself results in the same diagram.

Second, for the case when a diagram D has two or more available sorting moves, it suffices

to show that for any pair of available sorting moves s and t on D, any terminating sort

sequence starting with s will give the same resulting diagram as any terminating sort

sequence starting with t. As in the previous case, both s(D) and t(D) will be in a truth

region, ie. all terminating sort sequences S on s(D) will result in the same diagram, and

the same for t(D). In particular, if we can choose sorting sequences S on s(D) and T

on t(D) such that S(s(D)) = T (t(D)), the claim follows. There are two cases: if s and

t do not overlap, they commute and the trivial syzygy st = ts can be used; otherwise,

syzygies S s = T t will be needed for the argument.

Thus, for all induction steps below, we only need to verify that for the given diagram

D, any available sorting move on it does result in a diagram in the true region, and that

for any pair of available overlapping sorting moves s,t on D, there are specific syzygies

S(s) = T (t(D)) to substitute into the above argument.


We proceed to check these for all steps in our two dimensional induction. Also recall

a sorting move is either an R2- or a GF -sort. First, for the Base “column, N = 0 ,

any diagram with zero illegal intervals has no available GF -sorts.

Base case, (N , χ) = (0, 0) or (0, 1) With less than two crossings, a diagram has no

available R2-sort either, so S(D) = D is well-defined.

Induction, “χ ≤ c− 1” ⇒ “χ = c” Assume S is well-defined on all diagrams with χ ≤

c− 1 where c ≥ 2. The only possible available sorting moves on a diagram D with

(N , χ) = (0, c) are R2-sorts, and by remark 3.2.5, any R2-sort on D will result in

a diagram in the truth region “χ ≤ c− 2.” Also, up to orientation of the strands,

there are only two possible ways R2-sorts can overlap, and they are mirror images

of one another. Here is the syzygy for one of them; the other one is analogous.

R2 R2

Figure 3.6: Syzygy for overlapping R2-sorts. Each sorting move happens inside the

corresponding sorting sites, boxed by light dotted lines, in different diagrams.

Secondly, the Induction step “columns N ≤ N − 1” ⇒ “column N = N”.

Base case, “columns N ≤ N − 1” ⇒ “(N , χ) = (N,χmin(N)) ” A diagram D with

the minimum number of crossing χmin to make N ≥ 1 illegal intervals has only one

crossing in each maximal over or under interval, and so has no available R2-sorts.

But GF -sorts can be available and by remark 3.2.5 result in diagrams in a truth

region. Now, there is only one way two GF -sorts can overlap and here is a syzygy

between them:


GF

GF

GF

GF

Figure 3.7: Syzygy for overlapping GF -sorts. Each sorting move happens inside the

corresponding sorting sites, boxed by light dotted lines, in different diagrams.

Induction, columns “N ≤ N − 1” and “N = N,χ ≤ c− 1” and ⇒ “(N , χ) = (N, c)”

Assume S is well-defined on all diagrams with less than N illegal intervals, where

N ≥ 1, and all diagrams with N illegal intervals and less than c crossings where

c > χmin(N). Now, on a diagram D with (N , χ) = (N, c) both GF - and R2-

sorts can be available and by remark 3.2.5, both will result in a diagram in a truth

region. We also need syzygies for all ways of overlap of all sorting moves, R2-R2,

GF -GF , and GF -R2. The first two cases R2-R2 and GF -GF are the same as in


previous steps, with syzygies shown in figures 3.6 and 3.3.1. For the third case,

a GF -sort and an R2-sort can overlap in essentially two ways up to orientation of

strands, depending on whether the crossings in the R2-sorting site belong to the

maximal over or under interval in the GF -sorting site. Also, within each of these

overlap types, the R2-sort site can still vary. Here is the syzygy for the first way;

the one for the second is analogous:

GF

GF

R2

R2

Figure 3.8: Syzygy for overlapping GF and R2-sorts. Similar syzygies where either or

both gray thick bands on the sides of R2-sort site in the top most diagram are removed

from all diagrams hold.

Lemma 3.3.2. The sorting map S is well-defined under Reidemeister-II and III moves.

Proof. This follows directly from proposition 3.2.2 and the next lemma 3.3.3.

Lemma 3.3.3. The sorting map S is well-defined under finger-moves.

Proof. Since S is well-defined under choice of different terminating sort sequences on all

diagrams D ∈ T Dvf, if we can choose a sorting sequences on both sides Dl and Dr of

the finger-move such that they result in the same diagram, then S(Dl) = S(Dr). The

following syzygy suffices:


GF GFGF

F

Figure 3.9: Syzygy for overlapping GF -sort and finger move.

This completes our proof of the classification of the framed version of pure descending

virtual tangles, the first statements in theorems1.0.1, 1.0.3.


3.4 Classification of the Unframed Version: Adding

Reidemeister I

To prove the second statements of theorems1.0.1, 1.0.3 which classify the unframed ver-

sion of pure descending virtual tangles, which recall is the quotient of the framed version

by the Reidemeister-I relation, we only need to slightly modify the proof of the framed

version in the last sections 3.2,3.3. First, we add an extra sorting move, the R1-sort as

shown below in figure 3.4, to the definition (3.2.7) of the sorting map.

R1

+/-

R1 R1

PLANAR GAUSS

Figure 3.10: R1-sort.

Then, we show that the modified sorting map is still well-defined by adding R1-sort

to the two-dimensional induction argument in 3.2.10: we note that performing any R1-

sort will either decrease the number of illegal intervals N by 1 or not change it, and will

always decrease the number of crossings χ by 1, thus resulting in a diagram in the already

true region in the induction domain; and use the following two overlapping syzygies to

conclude that the choice to perform an R1-sort, an R2-sort, or a GF -sort at each stage

of the sorting does not affect the result.

R1

R1

PLANAR GAUSS

R1 R1

R2

+/-

-/+

+/-

R2

Figure 3.11: Syzygy between R1- and R2- sorting moves.


GF

R2's

R1

R1

Figure 3.12: Syzygy between R1- and GF - sorting moves.

3.5 Examples

Here are some examples of the sorting map applied to descending virtual long knots and

pure tangles.

1. Less trivial knot example which simplifies quickly with R1 but not so without.


deform1 23 5

4

31 2 4

5

GF1 2 34

5

6

7

8

9

10

11

deform

476 3 11 2105819

R211 9 5 10 4

1 2 3 4

5

Figure 3.13: Example of the sorting algorithm.

2. Two simple tangle examples. Note that the starting diagrams are also in “braid-

form,” i.e. that as Gauss diagrams, the chords do not need to intersect one another;


however, the canonical diagrams may not remain in braid-form.

+

+

1

2

3

1

2

3

+ +

1

2

3

+

+

+

-

GFdeform

+

1

2

3

GF-

+

1

2

3

-

+ -

deform+

1

2

3

- +

-

Chapter 4

Basis of Avf

In this section, we prove theorems 1.0.4, 1.0.5 on page 11, which give bases for two arrow

diagram algebras Avfb(↑1↑2 . . . ↑n) and Avf(↑1↑2 . . . ↑n) associated to descending virtual

tangles with open labelled components.

We first describe a general argument in section 4.1 and apply this general argument

to our specific spaces in section 4.3.

We will rely on the terminology for decorated chord diagram spaces set up in sec-

tion ?? on page ??. In particular, notice the distinction between relations and generating

relations, the definition of the add-chord operation θa.

4.1 Generalized Grobner Basis for Chord Diagram

Algebra

In this section, we give an answer to a special case of the following general question: find

a basis for a given quotient space A = D/I of a decorated chord diagram space D, where

I is a finitely generated and homogeneous ideal in D. Our main observation is that the

Grobner basis argument for associative algebras can be extended to general chord dia-

gram algebras by replacing the role of the associative multiplication in associate algebras

58

Chapter 4. Basis of Avf 59

by the unary CTD-multiplication operators θa. It turns out that our CD-algebras of

interest, Avfb(↑1↑2 . . . ↑n) and Avf(↑1↑2 . . . ↑n), admit such generalized Grobner bases.

Our main statement is Lemma 4.1.8, which follows after a restatement of infinite di-

mensional Gaussian elimination and a lemma on the generators of differences of spanning

vectors of an ideal I in a general chord diagram algebra.

First, a convention: .Let V be a vector space over a field K, B a basis and O a partial

ordering on B. For any vector v =∑m

k=1 akbk, where 0 = ak ∈ K, bk ∈ B, bi = bj if i = j,

we call the maximal basis elements in bi, if they exist, the leading basis element of v,

and the terms proportional to these the leading terms of v.

Lemma 4.1.1 (Infinite Dimensional Gaussian Elimination). Given a vector space V of

all finite linear combinations of elements in a countable basis B, and a subspace W which

is the span of a (possibly infinite) set SW := w1, w2, . . ., if there is a partial ordering

on B such that w.r.t. it

1. any descending chain in B is finite;

2. each vector w in the spanning set SW has a unique leading term, denoted aw lw

where aw ∈ K and lw ∈ B;

3. the difference of each pair of vectors wi, wj in the spanning set SW which have

proportional leading terms, i.e. cawilwi

= awjlwj

where c ∈ K, can be written in

terms of vectors in SW with strictly lower leading terms as follows:

(cawiwi − awj

wj) =m∑k=1

awkwk where lwk

< lwi∀k

and call the set of all leading basis elements L := lwi∈ B | wi ∈ SW.

Then any subset of SW consisting of vector representatives of the leading basis elements,

i.e. for each l ∈ L, choose one wi ∈ SW with lwi= l, is a basis of W, and B − L is a

basis of V/W.


Proof. This is elementary linear algebra. Use induction on the partial ordering for both

statements. Condition 1 and that each vector in V consists of only finite linear combina-

tions ensure all needed algorithm terminate.

Like the usual Grobner basis argument on (non-commutative) associative algebras,

the Grobner basis argument for CD below improves the above, which demands infinitely

many syzygies, by using the property of the algebraic structure that “common multiples”

of any two elements are generated by “minimal common multiples” to reduce the search

for syzygies at the generator rather than vector level.

In the following, G is the space spanned by all decorated chord diagrams of a (linearly-

extended) free or free order-preserving chord diagram algebra CD(S)⟨χ1, . . . , χn⟩ on a

skeleton S, and GS is the subset spanned by diagrams on a subskeleton S ≤ S. Recall

(and refer to page 13) that θa : GS −→ GS′is a unary multiplication operator on G

mapping diagrams on the skeleton S to S ′ where the subscript a encodes all information

about the multiplication.

Definition 4.1.2. Acommon multiple of any two diagrams G,G′ ∈ G is a diagram L ∈ G

such that L = θa(G) = θa′(G′) for some θa and θa′ CD-multiplication operators in G, or

equivalently a diagram L which includes both G and ′G as subdiagrams. We denote

the common multiple L along with the two factorizations by L(G,θa),(G′,θa′ ) can call it a

common multiple type.

Recall from section 2.1.5 that subdiagrams are considered up to “combinatorics,” i.e.

the boundary points of the subdiagram can slide without touching vertices, chord ends,

and other open ends of the same subdiagram. Since the two subdiagrams G and G′ are

independent of one another in L, the above leads to a slight ambiguity when considering

their “union” and “intersection” in L:


G G'

=

GG'

Figure 4.1: Two equivalent ways of drawing parts of the boundaries of the the subdia-

grams G and G′. Thus, subdiagram in the intersection on the right is “removable” or

“trivial.”

Definition 4.1.3. The union of the two subdiagrams G, G′ in the common multiple

type L(G,θa),(G′,θa′ ) is the minimal (w.r.t. diagram inclusion) subdiagram in L which

includes both G and G′. The intersection of G, G′ in L(G,θa),(G′,θa′ ) to be the maximal

subdiagram included in both G and G′, or equivalently a maximal common factor of the

subdiagrams G and G′ in L(G,θa),(G′,θa′ ) . An intersection is trivial if in an equivalent

way of depicting the subdiagrams G and G′ in L(G,θa),(G′,θa′ ), the literal intersection of

these two depicted images (not considered up to subdiagram equivalence) is empty.

We distinguish common multiple types L(G,θa),(G′,θa′ ) into the overlapping and non-

overlapping ones.

Definition 4.1.4 (Overlapping and Non-overlapping Common Multiple Types). A com-

mon multiple type L(G,θa),(G′,θa′ ) of two diagrams G and G′ is called overlapping if

the subdiagrams G and G′ in has a non-trivial intersection (as in definition 4.1.3) in

L(G,θa),(G′,θa′ ). It is non-overlapping otherwise, i.e. if the subdiagrams G and G’

has trivial or no intersection in L(G,θa),(G′,θa′ ), or equivalently, a non-overlapping com-

mon multiple type of G and G′ is one which is a multiple of the disjoint union (up to

reparametrization) of G and G′ with the subdiagrams G and G′ remembered,

Here are some examples:


θ (G)a

θ (G')a'

=

=

L

G

=

=

G'

G

G'

=

=

Figure 4.2: Some overlapping common multiples. (L) G and G′ are empty diagrams.

The intersection of G and G′ in both cases are non-trivial since in all ways of depicting

G and G′ in L, the intersection of the drawn images of G and G′ is non- empty. Both of

these are minimal.

=

=

θ (G)a

θ (G')a'

=

=

L

=

=

G

G'

G

G'

Figure 4.3: Some Non-overlapping common multiples. (L) G and G′ are empty diagrams.

Notice that L is non-overlapping since it has only a trivial intersection, i.e. there is one

way of depicting G and G′ in which the intersection of the drawn images of G and G′ is

empty. The diagram on the right is minimal and on the left is not.

This in part follows from the associativity axioms of the CD operations.

Proposition/Definition 4.1.5 (Minimal Common Multiple Types). A common multi-

ple type L(G,θa),(G′,θa′ ) of two diagrams G and G′ is minimal if the common multiple L

is equal to the union (as in 4.1.3) of the subdiagrams G and G′ in it. The set of minimal

common multiple types of G and G′ generate all common multiple types of G and G′ in

the sense that any common multiple type is the output of a CD multiplication operation

with a minimal common multiple type (the diagram along with the two factorization) as

input. We call a minimal overlapping common multiple type simply an overlap type. Any


overlap type of G and G′ can be constructed by finding a common subdiagram of any

diagrams G, G′ resulting from any allowed reparametrization (none for order-preserving

algebras) of G and G′ , identifying (or gluing) G and G′ on this common subdiagram,

and retaining the output along with the subdiagrams G and G′ in it if this output is a

chord diagram and if the intersection of G and G′ in the output has no subdiagram of

the form shown in figure 4.1. The set of all overlap types are finite.

Proof. All common multiples contain the union of G and G′ so generation is clear. The

construction of the set of minimal common multiple types is direct from the definition

and the set is finite because the diagrams have finite combinatorics.

If R := g1, . . . , gn | gi ∈ G in which each vector gi is linear combination of only

diagrams on the same skeleton, then the above lemma also gives a generating set of all

pairs of vectors θa(g), θa′(g′) with the same leading diagrams.

Proposition/Definition 4.1.6. Given R as above. The set of all pairs

θa(g), θa′(g′)

with the same leading diagrams are generated by the subset whose common leading

diagram L := θa(Lg) = θa′(Lg′) is a minimal common multiple of the leading diagrams

Lg and Lg′ of g and g′. We denote this generating set of vectors with common leading

diagrams MR. For each pair in MR, let δ(g,θa),(g′,θa′ ,g′) := cθa(g) − θa′(g′) where the

scalar c is such that the leading terms in the two generating relations cθa(g) and θa′(g′)

cancel one another. Furthermore, if (Lg, θa), (Lg′ , θa′) is an overlap type of the leading

diagrams Lg and Lg′ , then call (g, θa), (g′, θa′) an overlap type of the generating relations

g and g′

Definition 4.1.7. A partial ordering on the generating diagrams G is said to respect the

CD structure if for any G,G′ ∈ G

G ≤ G′ ⇐⇒ θa(G) ≤ θa(G′) ∀ CD operations θa


Lemma 4.1.8 (Grobner Bases for Decorated Chord Diagram Algebras). Given a linearly

extended free or free order-preserving general chord diagram algebra on a skeleton S, in

which GS is the space spanned by the set of decorated chord diagrams on the subskeleton

S ≤ S,and θa : GS −→ GS′are CD-multiplication operators. Let I be an ideal generated

by a finite set of relations R := g1, . . . gn each of which is a linear combination of only

diagrams on the same skeleton. Then if there exists a partial ordering O on the set of all

decorated chord diagrams that respects the CD-structure, such that w.r.t it:

1. any descending chain in G is finite;

2. each generating relation gi has a (well-defined) unique leading term, and call the

set of all leading diagrams L and the subset with skeleton S LS;

3. for each pair of generating relations gi, gj (i possibly equals j), and each overlap

type(θb, ri), (θb′ , gj) of gi, gj, there exists a (generating) syzygy

δ(gi,θb),(gj ,θb′ ) =m∑k=1

ck θak(gk) where ck ∈ K (4.1)

such that the leading diagram L(gi,θb),(gj ,θb′ ) which will be canceled on the L.H.S.

is a (well-defined) unique maximum among all the leading diagrams Lθak (gk)of the

relations θak(gk) on the R.H.S. in the syzygy.

Then for each skeleton S ≤ S, a basis of IS := I ∪ GS is any set of relations in which

exactly one relation has leading diagram L for each leading diagram L ∈ LS, and a basis

for the quotient GS/IS is the set of all decorated chord diagrams on skeleton S which is

not inLS.

Proof. We apply the lemma 4.1.1 with V = GS, W = IS and the spanning set SIS ofW to

be the set θa(gi)∪GS of all multiples of the generating relations g1, . . . , gn with skeleton

S. It suffices to show that for any two relations r = θa(gi), r′ = θb(gj) in the spanning

set SIS with the same leading diagrams, there exists a syzygy such that the difference


cr − r′ can be written as a linear combination of relations in S all with lower leading

terms (w.r.t. O). Now, since the partial ordering on G respects any CD-multiplication

operation θa, it suffices to show there are such syzygies for a set that generates all pairs of

relations with same leading diagrams, and corollary 4.1.6 gives MR as a finite generating

set.

Summarizing, we only need to show that for any pair (gi, θa), (gj, θb) in MR,

δ(gi,θa),(gj ,θb) can be written as linear combinations of generation relations with lead-

ing terms that are strictly smaller than L(gi,θa),(gj ,θb).

There are two cases: if (gi, θa), (gj, θb) is an overlap type, then we can use the

syzygy given in condition 3;

if (gi, θa), (gj, θb) is not an overlap type, then θa(gi) = θc(gi ⊔ Lgj) and θb(gj) =

θc(Lgi ⊔ gj) for some operation θc, and there is the trivial syzygy

θc(gi ⊔ gj)− θc(gi ⊔ gj) = 0

⇔ θc(agiLgi ⊔ gj)− θc(gi ⊔ agjLgj) = −θc(∑k

a(k)giD(k)

gi⊔ gj) + θc(gi ⊔

∑k

a(k)gjD(k)

gj)

⇔ c δ(gi,θa),(gj ,θb) = agiθb(gj)− agjθa(gi) =∑k

(−a(k)giθbk(gj) + a(k)gj

θck(gi))

where agi , a(k)gi ∈ K and D

(k)gi are defined by gi = agiLgi +

∑k a

(k)gi D

(k)gi , and more im-

portantly, by transitivity of the partial ordering, the common leading diagram on the

L.H.S., L(gi,θa),(gj ,θb) = θc(Lgi ⊔ Lgj), is strictly bigger than the leading diagrams of all

generating relations on the R.H.S. of the syzygy.

4.2 A Partial Ordering on CD algebras

Before applying the above argument to our specific general chord diagram algebras, let

us describe a general way of defining partial orderings on G.


First, counting the number of ways a given generating diagram g ∈ G can be embedded

in any generating diagram in G gives an ordering on G:

Definition 4.2.1. Let G1, G2, . . . Gk ∈ G be generating diagrams and NGi: G −→ Z≥0

the number of different embeddings of Gi in D, i.e.

NGi(D) :=| θa | D = θa(G) | .

Then define the partial ordering on G induced by the ordered set of functions (NG1 , NG2 . . . NGk)

by

G < G′ ⇔ ∃ 1 ≤ n ≤ k s.t. NGi(G) = NGi

(G′) ∀ i < n , and NGn(G) < NGn(G′)

Remark 4.2.2. Thus, appending NGk+1to the ordered set (Ng1 , Ng2 . . . Ngk) gives a more

refined partial ordering induced by the functions. Also, clearly, any descending chain

w.r.t. a partial ordering on G defined this way is finite.

To use the argument in lemma 4.1.8, we need a partial ordering O on G that respects

all CD operations θa but also needs to compare only certain subsets of G. Thus, one

way to define O is to restrict the ordering induced by (NG1 , NG2 . . . NGk) to only the

relevant pairs of generating diagrams in G (i.e. those needed in lemma 4.1.8), and check

that the ordering induced by (NG1 , NG2 . . . NGk) on these pairs is indeed preserved under

all CD operation θa. We rephrase the conditions on O in lemma 4.1.8 as conditions on

(NG1 , NG2 . . . NGk):

Lemma 4.2.3. A partial ordering O on G which satisfies all conditions required in

lemma 4.1.8 can be constructed from the ordering induced by an ordered set of functions

(NG1 , NG2 . . . NGk) if the set satisfies the following:

1. for each gj =∑

s asHs ∈ R, there is one special diagram Lgj = Hs among all

diagrams Hs such that for some 1 ≤ n ≤ k and for all θa,

∀ i < n NGi(θa(Lgj)) = NGi

(θa(D)) and NGn(θa(Lgj)) > NGn(θa(D))

where D is any of the diagrams Hs = Lgj ;


2. for at least one syzygy (assuming existence) for each overlap type (gi, θa), (gj, θb)

of any pair of generating relations gi, gj, there is some 1 ≤ n ≤ k such that for all

θa,

∀ i < n, NGi(θa(L(gi,θa),(gj ,θb))) = NGi

(θa(Lr)) and NGn(θa(L(gi,θa),(gj ,θb))) > NGn(θa(Lr))

where Lr is any of the leading diagrams other than L(gi,θa),(gj ,θb) appearing in the

syzygy.

Proof. All follows directly from definitions.

Furthermore, the following lemma gives finite sufficient conditions for when the or-

dering induced by a function NG between a pair of generating diagrams respects the CD

structure.

Lemma 4.2.4. Let NG be a function as in definition 4.2.1, and L,D be generating

diagrams with the same skeleton S. Let

ΩGθa(H) := |θa′ | (H, θa), (G, θa′) is an overlap type of G and H |

be the number of overlap types of generating diagrams G and H that have the same

common multiple θa(H). Then if for each θa with ΩGθa(D) > 0, ΩG

θa(D) ≤ ΩG

θa(L), then

NG(θa(D)) ≤ NG(θa(L)) ∀ θa. (4.2)

If, moreover, ΩGId(D) < ΩG

Id(L), the inequalities 4.2 are strict. On the other hand,

if ΩGθa(D) = 0 ⇒ ΩG

θa(L) = 0 for all θa, and for the finite number of θa’s such that

ΩGθa(D) > 0, ΩG

θa(D) = ΩG

θa(L), then equality in 4.2 holds for all θa.

Proof. For each θa, the number of embeddings of G into θa(L) and θa(D) where G does

not overlap with either L or D are equal, so it suffices to compare contributions to

NG(θa(L)) and NG(θa(D)) from only embeddings of G in which G overlaps at least one

of L and D. Now, for any generating diagram H and any θa (compatible with H), the


contribution to NG(θa(H)) from the embeddings of G in which G does overlap H is the

sum ∑θb

| θa′ | θa′ θb = θa | × ΩGθb(H)

where the sum is over all θb such that (H, θb), (G, θb′) is an overlap type of H and G

for some θb′ . The first factor in each summand does not depend whether θa is applied on

D or L, and the second factor is as controlled by the conditions in the statement. The

inequalities/equalities between NG(θa(L)) and NG(θa(D)) for all θa’s then follow from

simple enumeration using the given conditions.

4.3 Bases for Associated Arrow Diagram Algebras

Using the general argument above, we now prove theorems 1.0.4, 1.0.5 on page 11,

which bases for the arrow diagram algebras Avfb(↑1↑2 . . . ↑n) and Avf(↑1↑2 . . . ↑n). First,

we summarize the definition of Avfb(↑1↑2 . . . ↑n) and Avf(↑1↑2 . . . ↑n) from section ??:

Avfb := Dvf/Ivfb where Dvf are descending arrow diagrams and Ivfb is the ideal generated

by the descending 6T generating relation; and Avf is a further quotient: Avf := Dvf/Ivf

where Ivf is the ideal generated by both the descending 6T and descending XII gener-

ating relations. Or pictorially,

+

-

6T:

XII :

:=

:=

+

For convenience later on, we also add the following third order XII3-generating rela-

tion, which is a consequence of the XII-generating relation, to the set of defining gener-

ating relations of Ivf:


- := XII3 : - =

In more technical language,Avfb(↑1↑2 . . . ↑n) is the free orientation-preserving general

chord diagram algebra the skeleton S :=↑1↑2 . . . ↑n modulo the ideal generated (via the

orientation-preserving operations) by the 6T generating relation above, where the arrows

points points from an earlier segment to a later segment of the skeleton.

all diagrams in this section, we may omit the arrows for simplicity but

use the convention left skeleton segments are always understood to precede

right skeleton segments.

We will prove the following lemma which immediately implies theorems 1.0.4, 1.0.5.

Lemma 4.3.1 (Basis for Avfb(↑1↑2 . . . ↑n) and Avf(↑1↑2 . . . ↑n)). A basis of Avfb(↑1↑2

. . . ↑n) is the subset Dvf − Lvfb of descending arrow diagrams where Lvfb is the set of all

diagrams generated by illegal generating diagram L6T:

Figure 4.4: The illegal generating diagram L6T

A basis of Avf(↑1↑2 . . . ↑n) is the subset Dvf −Lvf of arrow diagrams where Lvfb is the

set of diagrams generated by L6T above and the following generating diagrams LXII , and

LXII3.

Figure 4.5: The illegal diagrams LXII (L) and LXII3 (R)

Proof. For each of Avfb(↑1↑2 . . . ↑n) and Avfb(↑1↑2 . . . ↑n), we use the general argument

in lemma 4.1.8 with the partial ordering on the generating diagrams constructed from an


ordered set of functions N := (NG1 , NG2 . . . NGk) as in section 4.2. For Avfb, N = (NHT ),

and forAvf, N = (NHT , NX , NX3), where the generating diagramsHT , X, X3 are defined

below in definition 4.3.2.

Then for each case, it suffices to show the following. First, the ordered set of function

N satisfies condition 1 in lemma 4.2.3 with g1, g2, . . . gn substituted in by the generating

relations that generate the ideal, i.e. 6T for Avfb and 6T,XII,XII3 for Avf; this is

shown for each case in lemma ?? below. Secondly, for each overlap type between each

pair of generating relations in R, there exists at least one syzygy as in equation 4.1 such

that (NG1 , NG2 . . . NGk) satisfies condition 2 in lemma 4.2.3; this is shown for each case

in lemma 4.3.7 below.

Leading Terms of 6T , XII and XII3

In this section, we define two specific orderings, the second one being a refinement of the

first, on the Gvf, and show that these orderings give well-defined unique leading terms

for the relations defining Avfb and Avf respectively as well as for syzygies among these

relations.

Definition 4.3.2. Let HT, X, and X3 be the following generating diagrams:

Figure 4.6: HT : the ”Head preceding tail” generating diagram. Notice the head and the

tail do not need to be immediately adjacent to one another.

Figure 4.7: X : an ”X” arrow pair generating diagram


‘

Figure 4.8: X3 : an ”arrow crossing a pair of arrows” generating diagram. For later

purposes, we call pair of arrows on the middle two skeleton segments the “consecutive

pair” of arrows and the remaining arrow the “single” arrow.

Let NHT , NX and NX3 be as in definition 4.2.1. Following section 4.2, define the

partial ordering Ovfb on Gvf to be the one induced by the function (NHT ), and the partial

ordering Ovf on Gvf to be the further refined partial ordering induced by the ordered set

of functions (NHT , NX , NX3).

Here are a few counting lemmas. In all of these (lemmas 4.3.3, 4.3.4, 4.3.5), let G

and G′ be two generating diagrams with the same number of chords on the n labeled

oriented strand skeleton.

Lemma 4.3.3. Let G and G′ be as above. For each i from 1 to n, both the number of

chords starting on all solid strands labeled i or bigger and the number of chords ending on

all solid strands labelled i or smaller are greater or equal for G than for G′ iff ΩHTθa

(G) ≥

ΩHTθa

(G′) for all θa such that θa(G′) is an overlapping L.C.M. of G and HT with a 1

chord overlap; and equality holds if those two numbers are equal for all i for G and G′.

Lemma 4.3.4. Let G and G′ be as above. Then, for each pair 1 ≤ i ≤ j ≤ n, the

sum of the number of chords that start on any skeleton segment labelled ≤ i and end on

any skeleton segment labeled ≥ j and the number of chords that start on any skeleton

segment labelled > i and end on any skeleton segment < j for the generating diagram G

is greater than or equal to that for G′ iff ΩXθa(G) ≥ ΩX

θa(G′) for all θa such that θa(G

′) is

an overlapping L.C.M. of G and X with a 1 chord overlap; and equality holds if those

two numbers are equal for all i, j for G and G′.


Lemma 4.3.5. Let G and G′ be as above. Then for each pair 1 ≤ i < j ≤ n the number

of chords that start on any strand labelled ≤ i and end on any strand labeled ≥ j and the

sum of the number of consecutive chord pairs, defined in figure 4.3.2, that start on any

strand labelled > i and end on any strand labeled < j in G is greater than or equal to

that in G′ iff ΩX3θa

(G) ≥ ΩX3θa

(G′) for all θa such that θa(G′) is an overlapping L.C.M. of

G and X3 with the overlapping chords being either the consecutive pairs of chords or the

single chord in X3 (see figure 4.3.2); and equality holds if those two numbers are equal

for all i, j for G and G′.

Lemma 4.3.6. Let L6T , LXII , LXII3 be as in figures 4.4, 4.5, and D6T , DXII , DXII3

be any of the generating diagrams other than L6T , LXII , LXII3 in the 6T -, XII-, and

XII3-generating relations respectively, then Ovfb defines L6T to be the unique leading

diagram of 6T-generating relations, and Ovf further gives LXII and LXII3 as the unique

leading diagrams for the XII- and XII3- generating relations, and these definitions of

leading diagrams respect the CD structure. In more details:

For 6T NHT (θa(L6T )) > NHT (θa(D6T )) ∀ θa;

For XII NHT (θa(LXII)) = NHT (θa(DXII)) ∀ θa, and NX(θa(LXII)) > NX(θa(DXII)) ∀ θa;

For XII3 NHT (θa(LXII3)) = NHT (θa(DXII3)) ∀ θa, NX(θa(LXII3)) = NX(θa(DXII3)) ∀ θa,

and NX3(θa(LXII3)) = NX3(θa(DXII3)) ∀ θa.

Proof. Recall from lemma 4.2.4 for G generating diagrams and g a generating relation,

we only need to show the set of inequalities NG(θa(Lg)) > NG(θa(Dg)) ∀ θa, it suffices

to show ΩGId(Lg) > ΩG

Id(Dg) and ΩGθa(Lg) ≥ ΩG

θa(Dg) for all θa such that θa(Dg) is an

overlapping L.C.M. of G and Dg; and similarly to show NG(θa(Lg)) = NG(θa(Dg)) ∀ θa,

it suffices to show the equalities between ΩGθa(Lg) and ΩG

θa(Dg) for any θa such that θa(Dg)

is an overlapping L.C.M. of G and Dg.

First, we show the three (in)-equalities pertaining to NHT . Clearly, HT can be

embedded into L6T , but not into any other diagramsD6T in the 6T -generating relation, so


ΩHTθId

(L6T ) = 1 > ΩHTθId

(D6T ) = 0; on the other hand, HT cannot be embedded into any

of the generating diagrams in the XII and XII3 generating relations, so ΩHTθId

(LXII) =

ΩHTθId

(DXII) = 0 and ΩHTθId

(LXII3) = ΩHTθId

(DXII3) = 0. Now, no other operations θa =

Id yields θa(G) as an overlapping L.C.M. of HT and G with a 2 chord overlap for

G any of the generating diagrams D6T in the 6T - and all of the ones in the XII- and

XII3- generating relations. Thus, we need to show the respective (in)-equalities between

ΩHTθa

(G) of respective generating diagrams G for only the operations θa which gives θa(G)

as an overlapping L.C.M. with a 1 chord overlap, but then a simple comparison of the

numbers of all chords starting on segment i or after and the numbers of chords ending on

segment i or before in the relevant generating diagrams and lemma 4.3.3 give all needed

(in)-equalities.

Secondly, we show the two (in)-equalities pertaining to NX . Arguing as above, the

generating diagram X embeds in one way into LXII , but not into DXII , and in two ways

into both LXII3 and DXII3, thus ΩXθId

(LXII) = 1 > ΩXθId

(DXII) = 0, and ΩXθId

(LXII3) =

ΩXθId

(DXII3) = 2. Now, again, no other operations θa = Id yields θa(G) as an overlapping

L.C.M. of X and G with a 2-chord overlap for G any of the generating diagrams DXII in

the XII- and all of the ones in the XII3- generating relations. Thus, we need to show

the respective (in)-equalities between ΩXθa(G) of the respective generating diagrams G for

only the operations θa which gives θa(G) as an overlapping L.C.M. with a 1 chord overlap,

but then a simple comparison of the numbers of all chords starting on segment i or after

and the numbers of chords ending on segment i or before in the relevant generating

diagrams and lemma 4.3.4 give the needed (in)-equalities.

Finally, we show the two (in)-equalities pertaining to NX3. Arguing as above, the gen-

erating diagramX3 embeds in one way into LXII3, but not intoDXII3, thus ΩX3θId

(LXII3) =

1 > ΩX3θId

(DXII3) = 0. Now, again, no other operations θa = Id yields θa(G) as an

overlapping L.C.M. of X3 and G with a 3-chord overlap or a 2 or 1-chord overlap one

chord from the consecutive chord pair for G any of the generating diagrams DXII in the



ΩX3θa

(G) of the respective generating diagrams G for only the operations θa which gives

θa(G) as an overlapping L.C.M. with a 1-chord overlap being the single chord in X3

or a 2-chord overlap being the consecutive pair in X3, but then a simple comparison

of the numbers of all chords starting on segment < i ending on ≥ j and the numbers

of consecutive chord-pairs starting on segment > i ending on segment ≤ j for all i, j

(not necessarily different) in the relevant generating diagrams and lemma 4.3.5 give the

needed (in)-equalities.

Syzygies and Their Maximum Leading Diagrams

Now that we have well-defined leading terms in the generating relations, we look for the

syzygies for each overlap type of each pair of generating relations and then check that the

overlap diagrams are the unique maxima among the leading diagrams of all generating

relations appearing in the syzygies.

Here are the overlap diagrams associated to each overlap type and also one syzygy

for each overlap type.

6T-6T: There is only one overlap type and its overlap diagram L6T−6T and the

associated δ6T−6T are:

Figure 4.9: δ6T−6T (L) ; L6T−6T (R)

-

where in L6T−6T the two chords on the right belongs to the leading term of a 6T-

generating relation and the two chords on the left belongs to the leading term of another

6T-generating relation. The following generating syzygy associated to this overlap type

gives δ6T−6T in terms of a linear combination of other 6T-generating relations:


= 06T-6T :

where the sum is over the 4 ways of placing the ends of the 6T-generating relator

and for each way the 3 ways of placing the extra chord with an end not on any segment

on which the 6T-generating relator has ended, and the brackets are the “commutator”

brackets as usual (see figure ??); thus, there are 24 terms in total.

XII-XII: There is only one overlap type and the overlap diagram LXII−XII and the

associated δXII−XII are as follow:

Figure 4.10: δXII−XII (L) ; LXII−XII (R)

-

The associated syzygy XII-XII is by construction as shown in figure 4.3.

XII3-XII3: There is no XII3-XII3 overlap.

6T-XII: There are a left and a right overlap types with respective overlap diagrams

L6T−XII(L) and L6T−XII(R) and associated δ6T−XII(L) and δ6T−XII(R) as follow:

Figure 4.11: δ6T−XII(L) (L) ; L6T−XII(L) (R)

-

Figure 4.12: δ6T−XII(R) (L) ; L6T−XII(R) (R)

-

The syzygies 6T-XII(L) and 6T-XII(R) associated to the respective overlap types are

respectively the equalities on the left and the right below:


+ = _ _

= +

L L.H.S. - MID. R R.H.S. - MID.= 0 = 06T-XII (L) 6T-XII (R): ::= :=

where the short-hand notations are as follow:

: +:= _ _

:= +

6T-XII3: Again, there are a left and a right overlap types with respective overlap

diagrams LXII3−6T and L6T−XII3 and associated δ6T−XII3(L) and δ6T−XII3(R) as follow:

Figure 4.13: δ6T−XII3(L) (L) ; L6T−XII3(L) (R)

-

Figure 4.14: δ6T−XII3(R) (L) ; L6T−XII3(R) (R)

-

The associated syzygy 6T-XII3(L) (resp. 6T-XII3(R)) is the sum of a multiple of the

6T-XII(L) (resp. 6T-XII(R)) syzygy and a trivial 6T-XII syzygy:

L 0 + = 0L 6T-XII3 (L) :

where the trivial 6T-XII syzygy is


:= -

+ - + + - -

0L

and the following generating relation appears twice and has been canceled:

and the following difference of generating relations has been rewritten:

- =

The right syzygy is exactly analogous and these syzygies are of the overlap types in

figure 4.12.

XII-XII3: There is only one overlap type and its overlap diagram LXII−XII3 and

associated δXII−XII3 are as follow:

Figure 4.15: δXII−XII3 (L) ; LXII−XII3 (R)

-

the associated syzygy XII-XII3 is:

- - - = 0XII-XII3 :

Lemma 4.3.7. Each of the overlap diagrams L6T−6T , LXII−XII , L6T−XII(L), L6T−XII(R),

L6T−XII3(L), L6T−XII3(R), LXIIXII3, in figures 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, and 4.15

respectively, is the unique maximal generating diagram among the leading diagrams of all

the generating relations in the respective syzygy w.r.t to the partial ordering O(NHT ,NX ,NX3)


induced by (NHT , NX , NX3). In more details, first, in any syzygy S, w.r.t. O(NHT ), the

leading diagrams LHT,iS listed below are maximal among all leading diagrams in S and the

maximality respects the general chord diagram algebra structure, i.e.

NHT (θa(LHT,iS )) = NHT (θa(L

HT,jS )) ∀i, j ∀ θa and NHT (θa(L

HT,iS )) > NHT (θa(D

HT,iS )) ∀i ∀ θa

where D(HT )S is any leading diagram not LHT,i

S in S:

6T-6T L6T−6T in figure 4.9;

6T-XII(L/R) L6T−XII(L) ( resp. L6T−XII(R)) in figure 4.11 ( resp. 4.12) and LHT6T−XII(L)

( resp. LHT6T−XII(R)) below:

L 6T-XII (L)

HT

L =

L 6T-XII (R)

HT

L =

6T-XII3(L/R) L6T−XII3(L) ( resp. L6T−XII3(R)) in figure 4.13 ( resp. 4.14) and LHT6T−XII3(L)

( resp. LHT6T−XII3(R)) below:

L 6T-XII3 (L)

HT

L =

L 6T-XII3 (R)

HT

L =

XII-XII, XII-XII3 all leading diagrams in the syzygies;

secondly, for each syzygy with non-unique maximal leading diagrams w.r.t. to O(NHT ),

the following leading diagrams are maximal w.r.t. to O(NHT ,NX):

6T-XII(L), 6T-XII(R), XII-XII L6T−XII(L), L6T−XII(R)), and LXII−XII in figures 4.11,

4.12 and 4.10respectively;

6T-XII3(L/R), XII-XII3 all maximal leading diagrams w.r.t. to O(NHT ) (as above);


and finally for each syzygy with non-unique maximal leading diagrams w.r.t. to

O(NHT ,NX), the following leading diagrams are maximal w.r.t. O(NHT ,NX ,NX3):

6T-XII3(L), 6T-XII3(R), XII-XII3 L6T−XII3(L), L6T−XII3(R), and LXII−XII3 in fig-

ures 4.11, 4.12 and 4.15 respectively.

Proof. We argue as in lemma 4.3.7 and reduce all needed the (in)-equalities in NHT , NX ,

NX3 to the respective (in)-equalities in Ω’s given in lemma ??.

First, w.r.t. to O(NHT ). A simply enumeration shows that the numbers of embeddings

of HT into the generating diagrams L6T−6T , L6T−XII(L/R), LHT6T−XII(L/R), L6T−XII3(L/R),

and LHT6T−XII3(L/R) are strictly greater than those for any other leading diagrams in the

respective syzygies, and there is no possible embedding of HT into any leading diagrams

in the XII − XII-, and XII − XII3- syzygies. Now, as in lemma 4.3.7, for any G

which is a leading generating diagram in any of the syzygies, no operations θa = Id

yields θa(G) as an overlapping L.C.M. between HT and G with a 2 chord overlap, and

so as in lemma 4.3.7, all needed (in)-equalities involving different Ω’s follow from simple

comparison of the number of all chords starting on segment i or after and the numbers

of chords ending on segment i or before between different generating leading diagrams

for all i’s and lemma 4.3.3.

Second, the maximal leading diagrams w.r.t. to the refined ordering O(NHT ,NX).

Among the Arguing as above, a simple enumeration shows that the numbers of em-

beddings of X into the generating diagrams L6T−XII(L/R), L6T−XII3(L/R), LXII−XII are

strictly greater than those for LHT6T−XII(L/R), L

HT6T−XII3(L/R), and other leading diagrams

in XII − XII-syzygy respectively, and is equal to 5 for all leading diagrams in the

XII − XII3-syzygy. As above, no other operations θa = Id on any of the relevant

generating diagrams yields as an overlapping L.C.M. of X and G with a 2-chord overlap.

Thus, we need to show the respective (in)-equalities between ΩXθa(G) of the respective

generating diagrams G for only the operations θa which gives θa(G) as an overlapping

L.C.M. with a 1 chord overlap, but then a simple comparison of the numbers of all


chords starting on strand ≥ i ending on strand leqj and lemma 4.3.4 give the needed

(in)-equalities.

Finally, we show the two (in)-equalities pertaining to NX3. Arguing as above, the

generating diagramX3 embeds in one way into L6T−XII3(L/R), but not into LHT6T−XII3(L/R),

and in two ways into LXII−XII3 but just in one way into the other leading diagrams in

XII − XII3-syzygy; thus ΩX3θId

(L6T−XII3(L/R)) = 1 > ΩX3θId

(LHT6T−XII3(L/R)) = 0 and

ΩX3θId

(LXII−XII3) = 2 > ΩX3θId

(DXII−XII3) = 1. Now, again, no operations θa = Id on

L6T−XII3(L/R), LHT6T−XII3(L/R) or any of the leading diagrams in theXII−XII3 generating

syzygy yields an overlapping L.C.M. with X3 with a 3-chord overlap. Furthermore, on

any of the leading diagrams in the XII−XII3 generating syzygy, no operations θa = Id

yields an overlapping L.C.M. with X3 with a 2 or 1-chord overlap with exactly one

chord from the consecutive chord pair in X3 overlapping, there is an obvious bijection

between all overlap L.C.M.s of this kind ofX3 with L6T−XII3(L/R) and with LHT6T−XII3(L/R).

Thus, we need to show the respective (in)-equalities between ΩX3θa

(G) of the respective

generating diagrams G for only the operations θa which gives θa(G) as an overlapping

L.C.M. with a 1-chord overlap being the single chord in X3 or a 2-chord overlap being

the consecutive pair in X3, but then a simple comparison of the numbers of all chords

starting on segment < i ending on ≥ j and the numbers of consecutive chord-pairs

starting on segment > i ending on segment ≤ j for all i, j (not necessarily different) in

the relevant generating diagrams and lemma 4.3.5 give the needed (in)-equalities.

Appendix A

Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5

Here are the counting arguments for lemmas 4.3.3, 4.3.4, 4.3.5. In all of these, let G and

G′ be two generating diagrams with the same number of chords on the n labeled oriented

strand skeleton.

Lemma A.0.8. Let G and G′ be as above. For each i from 1 to n, both the number of

chords starting on all solid strands labeled i or bigger and the number of chords ending on

all solid strands labelled i or smaller are greater or equal for G than for G′ iff ΩHTθa

(G) ≥

ΩHTθa

(G′) for all θa such that θa(G′) is an overlapping L.C.M. of G and HT with a 1

chord overlap; and equality holds if those two numbers are equal for all i for G and G′.

Proof. Any CD operation θa such that θa(G) is an overlapping L.C.M. of HT and G with

a 1 chord overlap has to result in a diagram in which two solid segments with a chord

across them are embedded onto the open skeleton intervals of G, and the “added chord”

from θa has to belong to HT in any decomposition of the L.C.M. θa(G) into θa′(HT ).

Then, for each such θa, ΩHTθa

(G) is the sum of the number of chords that start on any

skeleton segments after the ending skeleton segment of the added chord and the number

of chords that end on any skeleton segments before the starting segment of the added

chord.

Lemma A.0.9. Let G and G′ be as above. Then, for each pair 1 ≤ i ≤ j ≤ n, the

81

Appendix A. Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5 82

sum of the number of chords that start on any skeleton segment labelled ≤ i and end on

any skeleton segment labeled ≥ j and the number of chords that start on any skeleton

segment labelled > i and end on any skeleton segment < j for the generating diagram G

is greater than or equal to that for G′ iff ΩXθa(G) ≥ ΩX

θa(G′) for all θa such that θa(G

′) is

an overlapping L.C.M. of G and X with a 1 chord overlap; and equality holds if those

two numbers are equal for all i, j for G and G′.

Proof. Any CD operation θa such that θa(G) is an overlapping L.C.M. of X and G with

a 1 chord overlap has to result in a diagram in which two solid segments with a chord

across them are embedded onto the open skeleton intervals of G, and the “added chord”

from θa has to belong to X in any decomposition of the L.C.M. θa(G) into θa′(X). Then,

for each such θa, ΩXθa(G) is the sum of the number of all chords that start on any skeleton

segment before the starting segment of the added chord and end on any skeleton segment

after the ending segment of the added chord and the number of all chords that start

on any skeleton segment after the starting segment of the added chord and end on any

skeleton segment before the ending segment of the added chord.

Lemma A.0.10. Let G and G′ be as above. Then for each pair 1 ≤ i < j ≤ n the

number of chords that start on any strand labelled ≤ i and end on any strand labeled ≥ j

and the sum of the number of consecutive chord pairs, defined in figure 4.3.2, that start

on any strand labelled > i and end on any strand labeled < j in G is greater than or equal

to that in G′ iff ΩX3θa

(G) ≥ ΩX3θa

(G′) for all θa such that θa(G′) is an overlapping L.C.M.

of G and X3 with the overlapping chords being either the consecutive pairs of chords or

the single chord in X3 (see figure 4.3.2); and equality holds if those two numbers are

equal for all i, j for G and G′.

Proof. Any CD operation θa such that θa(G) is an overlapping L.C.M. of X3 and G with

a 2 chord overlap being the consecutive pairs of chord in X3 has to result in a diagram in

which two solid segments with a chord across them are embedded onto the open skeleton


intervals (i, i+1), (j, j +1) of G, and the “added chord” from θa has to belong to X3 in

any decomposition of the L.C.M. θa(G) into θa′(X). Then, for each such θa, ΩX3θa

(G) is

the sum of the number of all consecutive chord-pairs that start on any skeleton segment

after the starting segment of the added chord and end on any skeleton segment before

the ending segment of the added chord. Similarly, any CD operation θa such that θa(G)

is an overlapping L.C.M. of X3 with a 1 chord overlap being the single chord in X3 has

to result in a diagram in which two solid segments with a consecutive chord-pair across

them are embedded onto the open skeleton intervals (i, i + 1), (j, j + 1) of G, and the

“added chord pair” from θa has to belong to X3 in any decomposition of the L.C.M.

θa(G) into θa′(X). Then, for each such θa, ΩX3θa

(G) is the sum of the number of all chords

that start on any skeleton segment before the starting segment of the added chord and

end on any skeleton segment after the ending segment of the added chord.

Lemma A.0.11. Let L6T , LXII , LXII3 be as in figures 4.4, 4.5, and D6T , DXII , DXII3

be any of the generating diagrams other than L6T , LXII , LXII3 in the 6T -, XII-, and

XII3-generating relations respectively, then Ovfb defines L6T to be the unique leading

diagram of 6T-generating relations, and Ovf further gives LXII and LXII3 as the unique

leading diagrams for the XII- and XII3- generating relations, and these definitions of

leading diagrams respect the CD structure. In more details:

For 6T NHT (θa(L6T )) > NHT (θa(D6T )) ∀ θa;

For XII NHT (θa(LXII)) = NHT (θa(DXII)) ∀ θa, and NX(θa(LXII)) > NX(θa(DXII)) ∀ θa;

For XII3 NHT (θa(LXII3)) = NHT (θa(DXII3)) ∀ θa, NX(θa(LXII3)) = NX(θa(DXII3)) ∀ θa,

and NX3(θa(LXII3)) = NX3(θa(DXII3)) ∀ θa.

Proof. Recall from lemma 4.2.4 for G generating diagrams and g a generating relation,

we only need to show the set of inequalities NG(θa(Lg)) > NG(θa(Dg)) ∀ θa, it suffices

to show ΩGId(Lg) > ΩG

Id(Dg) and ΩGθa(Lg) ≥ ΩG

θa(Dg) for all θa such that θa(Dg) is an


overlapping L.C.M. of G and Dg; and similarly to show NG(θa(Lg)) = NG(θa(Dg)) ∀ θa,

it suffices to show the equalities between ΩGθa(Lg) and ΩG

θa(Dg) for any θa such that θa(Dg)

is an overlapping L.C.M. of G and Dg.

First, we show the three (in)-equalities pertaining to NHT . Clearly, HT can be

embedded into L6T , but not into any other diagramsD6T in the 6T -generating relation, so

ΩHTθId

(L6T ) = 1 > ΩHTθId

(D6T ) = 0; on the other hand, HT cannot be embedded into any

of the generating diagrams in the XII and XII3 generating relations, so ΩHTθId

(LXII) =

ΩHTθId

(DXII) = 0 and ΩHTθId

(LXII3) = ΩHTθId

(DXII3) = 0. Now, no other operations θa =

Id yields θa(G) as an overlapping L.C.M. of HT and G with a 2 chord overlap for

G any of the generating diagrams D6T in the 6T - and all of the ones in the XII- and


ΩHTθa

(G) of respective generating diagrams G for only the operations θa which gives θa(G)

as an overlapping L.C.M. with a 1 chord overlap, but then a simple comparison of the

numbers of all chords starting on segment i or after and the numbers of chords ending on

segment i or before in the relevant generating diagrams and lemma 4.3.3 give all needed

(in)-equalities.

Secondly, we show the two (in)-equalities pertaining to NX . Arguing as above, the

generating diagram X embeds in one way into LXII , but not into DXII , and in two ways

into both LXII3 and DXII3, thus ΩXθId

(LXII) = 1 > ΩXθId

(DXII) = 0, and ΩXθId

(LXII3) =

ΩXθId

(DXII3) = 2. Now, again, no other operations θa = Id yields θa(G) as an overlapping

L.C.M. of X and G with a 2-chord overlap for G any of the generating diagrams DXII in

the XII- and all of the ones in the XII3- generating relations. Thus, we need to show

the respective (in)-equalities between ΩXθa(G) of the respective generating diagrams G for

only the operations θa which gives θa(G) as an overlapping L.C.M. with a 1 chord overlap,

but then a simple comparison of the numbers of all chords starting on segment i or after

and the numbers of chords ending on segment i or before in the relevant generating

diagrams and lemma 4.3.4 give the needed (in)-equalities.


Finally, we show the two (in)-equalities pertaining to NX3. Arguing as above, the gen-

erating diagramX3 embeds in one way into LXII3, but not intoDXII3, thus ΩX3θId

(LXII3) =

1 > ΩX3θId

(DXII3) = 0. Now, again, no other operations θa = Id yields θa(G) as an

overlapping L.C.M. of X3 and G with a 3-chord overlap or a 2 or 1-chord overlap one

chord from the consecutive chord pair for G any of the generating diagrams DXII in the


ΩX3θa

(G) of the respective generating diagrams G for only the operations θa which gives

θa(G) as an overlapping L.C.M. with a 1-chord overlap being the single chord in X3

or a 2-chord overlap being the consecutive pair in X3, but then a simple comparison

of the numbers of all chords starting on segment < i ending on ≥ j and the numbers

of consecutive chord-pairs starting on segment > i ending on segment ≤ j for all i, j

(not necessarily different) in the relevant generating diagrams and lemma 4.3.5 give the

needed (in)-equalities.

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