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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]
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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.
Chapter 1. Introduction 5
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.
Chapter 1. Introduction 6
...
...
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
Chapter 1. Introduction 7
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:
Chapter 1. Introduction 8
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.
Chapter 1. Introduction 9
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
Chapter 1. Introduction 10
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.
Chapter 1. Introduction 11
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.
Chapter 1. Introduction 12
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
Chapter 2. Preliminaries 15
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:
Chapter 2. Preliminaries 16
, 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.
Chapter 2. Preliminaries 17
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
Chapter 2. Preliminaries 18
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
Chapter 2. Preliminaries 19
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.
Chapter 2. Preliminaries 20
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
Chapter 2. Preliminaries 21
: 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
Chapter 2. Preliminaries 22
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
Chapter 2. Preliminaries 23
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
Chapter 2. Preliminaries 24
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
Chapter 2. Preliminaries 25
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.
Chapter 2. Preliminaries 26
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
Chapter 2. Preliminaries 27
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.
Chapter 2. Preliminaries 28
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
Chapter 2. Preliminaries 29
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).
Chapter 2. Preliminaries 30
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
Chapter 2. Preliminaries 31
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
Chapter 2. Preliminaries 32
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.
Chapter 2. Preliminaries 33
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
Chapter 2. Preliminaries 34
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:
+:= :=_ _ _
:=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 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.
Chapter 2. Preliminaries 35
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.
Chapter 2. Preliminaries 36
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
Chapter 2. Preliminaries 37
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):
+:= :=_ _ _
:=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 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.
Chapter 2. Preliminaries 38
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
Chapter 2. Preliminaries 39
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.
Chapter 2. Preliminaries 40
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:
Chapter 3. Classification of Pure Descending Virtual Tangles 43
...
...
...
...
...
...
...
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
Chapter 3. Classification of Pure Descending Virtual Tangles 44
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
Chapter 3. Classification of Pure Descending Virtual Tangles 45
((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.
Chapter 3. Classification of Pure Descending Virtual Tangles 46
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 47
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
Chapter 3. Classification of Pure Descending Virtual Tangles 48
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
Chapter 3. Classification of Pure Descending Virtual Tangles 49
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 50
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:
Chapter 3. Classification of Pure Descending Virtual Tangles 51
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
Chapter 3. Classification of Pure Descending Virtual Tangles 52
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:
Chapter 3. Classification of Pure Descending Virtual Tangles 53
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 54
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 55
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 56
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;
Chapter 3. Classification of Pure Descending Virtual Tangles 57
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.
Chapter 4. Basis of Avf 60
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:
Chapter 4. Basis of Avf 61
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:
Chapter 4. Basis of Avf 62
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
Chapter 4. Basis of Avf 63
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
Chapter 4. Basis of Avf 64
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
Chapter 4. Basis of Avf 65
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.
Chapter 4. Basis of Avf 66
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 ;
Chapter 4. Basis of Avf 67
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
Chapter 4. Basis of Avf 68
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:
Chapter 4. Basis of Avf 69
- := 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
Chapter 4. Basis of Avf 70
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
Chapter 4. Basis of Avf 71
‘
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′.
Chapter 4. Basis of Avf 72
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
Chapter 4. Basis of Avf 73
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
Chapter 4. Basis of Avf 74
XII3- generating relations. 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.
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:
Chapter 4. Basis of Avf 75
= 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:
Chapter 4. Basis of Avf 76
+ = _ _
= +
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
Chapter 4. Basis of Avf 77
:= -
+ - + + - -
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)
Chapter 4. Basis of Avf 78
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);
Chapter 4. Basis of Avf 79
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
Chapter 4. Basis of Avf 80
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
Appendix A. Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5 83
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
Appendix A. Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5 84
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.
Appendix A. Proofs of Lemmas 4.3.3, 4.3.4, 4.3.5 85
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
XII3- generating relations. 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.
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