Flat Virtual Pure Tangles - University of Toronto T-Space · virtual tangles whose skeleton is an ordered union of strands (in particular no closed loops) ﬂat virtual pure tangles

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 of interest because its finite-type invariant theory is potentially a topolog-

ical 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 cross-

ings, 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. The classification of the former can be used as an invariant on

virtual tangles with no closed components and virtual braids. In a subsequent paper, we

will show that the infinitesimal algebras are the target spaces of any universal finite-type

invariants on the respective variants of the flat virtual tangles.

ii

Acknowledgements

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, care, and countless hours of

inspiring discussion. I also thank him for being transparent with his thoughts through-

out research, for showing me much beauty, and showing me what it means to pursue a

problem until it becomes simple.

I am very grateful for the generous nurturing and encouragements that other profes-

sors and mentors have shown me, in particular Vassily Manturov, Louis Kauffman, and

Joel Kamnitzer.

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 un-

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

I first learned about flat virtual knots in a lecture by Vassily Manturov in the Trieste

summer school on knot theory.

Finally, I thank my family, Michael, and my friends for loving me throughout my life.

iii

Contents

1 Introduction 1

2 Preliminaries 13

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

2.1.1 Free General Chord Diagrams Algebra . . . . . . . . . . . . . . . 13

2.1.2 Crossings as Chords: Another Presentation of Chord Diagrams . . 20

2.1.3 Remarks on the Chord Diagram Algebra Operations . . . . . . . 21

2.1.4 Subdiagrams, Superdiagram, Embedded Subdiagrams . . . . . . . 25

2.1.5 Chord Diagram Subalgebras and Quotient Algebras . . . . . . . . 28

2.2 Virtual Pure Tangles, Flat Virtual Pure Tangles,and Their Variants . . . 30

2.2.1 Subsets of Reidemeister Moves . . . . . . . . . . . . . . . . . . . . 32

3 Classification of Pure Descending Virtual Tangles 36

3.1 Generic Diagrams of Pure Descending Virtual Tangles . . . . . . . . . . . 36

3.2 The Sorting Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Sorting map is well-defined . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Classification of the Unframed Version: Adding Reidemeister I . . . . . . 49

3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Preliminaries: Minimal Common Multiples of Chord Diagrams 54

iv

4.1 On Partial Ordering Induced by Containment . . . . . . . . . . . . . . . 54

4.2 Common Multiples, and Common Factors . . . . . . . . . . . . . . . . . 59

5 Preliminaries: the Associated Graded Spaces 64

5.1 Graded Spaces Associated to Filtered Vector Spaces . . . . . . . . . . . . 64

5.2 Linear Extension of Chord Diagram Algebras . . . . . . . . . . . . . . . 65

5.3 Brief Introduction to Finite Type Invariant Theory . . . . . . . . . . . . 66

5.4 Associated Graded Spaces of Free Chord Diagram algebras . . . . . . . . 68

5.5 The Associated Graded Spaces of Chord Diagram Algebra . . . . . . . . 70

5.6 The Associated Graded Spaces of vPT , fPT , and dPT . . . . . . . . . 71

6 Proof of Bases of Afb and Af 75

6.1 Grobner Argument for Chord Diagram Algebras . . . . . . . . . . . . . . 75

6.2 Partial Orderings on Chord Diagrams . . . . . . . . . . . . . . . . . . . . 79

6.3 Restatement of Bases for Arrow Diagram Algebras . . . . . . . . . . . . 84

6.4 Some Lemmas on Counting Embeddings of HT , X, X3 . . . . . . . . . . 86

6.5 Grobner Argument applied to Afb . . . . . . . . . . . . . . . . . . . . . . 90

6.6 Grobner Argument applied to Af . . . . . . . . . . . . . . . . . . . . . . 96

6.6.1 Well-definedness of Leading Terms of Relations . . . . . . . . . . 96

6.6.2 Enumeration of Overlap Types and Syzygies . . . . . . . . . . . . 97

6.6.3 Well-Definedness of Maximum Leading Diagrams of Syzygies . . . 100

Bibliography 102

v

Chapter 1

Introduction

We study virtual knots because they are a natural generalization of classical knots into

which classical knots inject, and more interestingly, because the R-matrix invariants on

classical knots extend naturally to virtual knots (or at least a variant of them).

We will define virtual knots by first recalling the definition of classical knots and

generalize from it. Classical knots can be defined combinatorially as knots diagrams

modulo Reidemeister moves. Knot diagrams are planar directed graphs with “crossings”

as vertices. Crossings are special tetravalent vertices whose half-edges are cyclically-

ordered and directed such that opposite pairs are “in-out” pairs. The crossings have

exactly the combinatorial information to be represented as follows:

+: -:

The Reidemeister moves are local planar graph equivalence relations shown below

where each skeleton strand can be oriented either way.

1

Chapter 1. Introduction 2

R2 R3R1

Virtual knots have the same definition except with word “planar” omitted, i.e. virtual

knot diagrams are (not-necessarily planar) graphs with crossings as vertices, and virtual

knots are equivalence classes of virtual knot diagrams under the Reidemeister relations as

local graph relations. Now, when not-necessarily planar graphs are drawn (or immersed)

on the plane, transverse intersections of edges of the graph may occur. These are not

vertices of the graph, but rather artifacts of drawing a non-planar graph on the plane,

and are called “virtual crossings.” Here is a virtual knot diagram drawn in two different

ways on the plane where the real crossings are circled and the other intersections are

virtual crossings:

=as v-knot

diagram

Similarly, the strand between any two crossings in a Reidemeister relation may inter-

sect other strands in the virtual knot diagram when drawn on the plane and have virtual

crossings on them.

A natural question arises: how much bigger are virtual knots than classical knots?

This leads to the consideration of the quotient of virtual knots by the crossing-flip relation

which equates the (real) positive and negative crossings:

Flat


in which all classical knots are equivalent to the unknot. This quotient is called flat

virtual knots.

The subject of this thesis is flat virtual long knots, where “long” simply refers to the

skeleton of the knot being a long line, and the “skeleton” is the union of lines and/or

circles obtained from tracing the knot diagram along the direction of the edges, across

paired half-edges at crossings and forgetting the crossings.

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

of flat virtual long knots, where “framed” means the Reidemeister 1 relation is not

imposed and “unframed” means otherwise. It can be shown that flat virtual long knots

are equivalent to descending virtual long knots, the subset of virtual knots with only

crossings whose over strand is earlier in the skeleton than the under strand w.r.t. to the

orientation of the skeleton. We give a canonical representative for each equivalent class

of descending virtual long knot diagrams under the Reidemeister moves.

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

Natan). Framed descending virtual long knots Kvf are in bijection with the set of canoni-

cal diagrams C1. A canonical diagram is a descending virtual long knot diagram whose

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

in, and after which as the under strand, and does not contain bigons bounded by opposite

signed crossings. An example is given in figure 1.1 and the general form is shown in

figure 1.2.

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 signed permutation satisfies

the extra condition that the image of pairs of consecutive numbers are not pairs of con-

secutive numbers with opposite signs, i.e. not ((j,∓), (j + 1,±)), or ((j + 1,±), (j,∓))

for all j < n.


Long unframed flat virtual knots Kvf are in bijection with the subset of C1 with no “R1

kinks,” also shown in 1.2. See figure 1 for a list of canonical diagrams up to three

crossings.

- + + + - σ(2) = (1 , + )

σ(1) = (3 , - ) σ :

σ(3) = ( 4, + )

σ(4) = ( 2, + )

σ(5) = (5 , - )

43 5 1 2 1 2 3 4 5 2 4 1 3 5

- - + + +

PLANAR GAUSSREDUCED SIGNED

PERMUTATION

Figure 1.1. Example of the canonical form of a descending virtual long knot.

There is a point on the skeleton before which it is the over strand in all crossings it

participate in and after which it is under.

...

...

PLANAR GAUSS

σ

... ...

where if R-1 imposed also: where

+/- +/-

-/+ -/+

if R-1 imposed also:

+/-

Ɛ1

Ɛ2

Ɛk

Figure 1.2. General form of the canonical diagrams of descending virtual long

knots, characterized by the existence of a point on the skeleton before which it is

the over strand in all crossings it participates in and after which it is under, and

the exclusion of the bigons and “R1-kinks” as well for the unframed version. In

the Gauss diagrams on the right, the ϵ’s are signs of crossings, and the box with σ

denotes a permutation of the arrows so that the incoming arrows are permuted by

σ within the box and emerge on the other side permuted.


R1 R1

R1 R1

Figure 1.3. List of canonical diagrams in Gauss diagram form of framed flat virtual

long knots up to three crossings. For the unframed version, exclude the diagrams

with R1 below them. Chords in the same diagram with dots on their left are required

to have the same signs; all others can be either + or −. Thus, the first diagram

in the second row represents 2 × 2 different canonical diagrams with different sign

arrangements.

The above result can be generalized easily to the multi-strand case. We call flat

virtual tangles whose skeleton is an ordered union of strands (in particular no closed

loops) flat virtual pure tangles. Similar to the long knot case, these are again equivalent

to descending virtual pure tangles.

Theorem 1.0.2 (Classification of Descending Virtual Pure Tangles). Framed descending

virtual pure tangles of n strands are in bijection with the set of canonical diagrams Cn,

which are characterized by the same two conditions as in the one strand case in theo-

rem 1.0.1 but applied to all n strands. Unframed descending virtual pure tangles are in

bijection with the subset of Cn with no “R-I kinks”. See figure 1.4 for an example, and

figure 1.5 for a partial list of canonical diagrams of descending virtual pure tangles on

two strands up to two crossings.


1 2 3 4

1' 2'

1 3 4 3 1

1' 2' 1' 2 4 2'

+ + +

- -

-

+

+

+

- -

-

PLANAR GAUSS

2

Figure 1.4. The canonical diagram of a framed descending virtual pure tangle on

two strands. On each skeleton strand, there is a point before which it is over in

all crossings and after which it is under in all crossings, and the diagram contains

no bigons bounded by opposite signed crossings. Since it also does not contain

any R1-kinks, it is a canonical diagram also of an unframed descending virtual pure

tangle.

R1 R1

Figure 1.5. List of canonical diagrams in Gauss diagram form up to two crossings

of framed descending virtual pure tangle on two strands with at least one crossing

between the two strands. For the unframed version, exclude the diagrams with R1

below them. Notice the top and bottom strands are distinguishable since the strands

are ordered. All dotted chords in the same diagram are required to have the same

signs, and all other are signed in all ways possible.

The proofs of both theorems are similar and amount to showing that a well-defined

sorting map exists. For an example of the sorting, see figure 1.

This classification can be use as an invariant on virtual long knots and pure tangles,

as well as virtual braids. If flat virtual braids inject into the pure tangles, then we have


:= :=_ _

Figure 1.6. Semi-virtual crossings corresponding to the positive or negative crossing,

defined as the formal difference between the corresponding real crossing and its

skeleton. The intersections of strands in the last terms are virtual crossings.

also obtained their classification.

The second part of this thesis gives bases for the associated graded spaces of flat virtual

long knots. These spaces are interesting because they are the target spaces for “universal

finite-type invariants”[BN2] [Pol] [GPV]. The filtration that gives these graded spaces

are the one the number of “semi-virtual” crossings, symbols for the formal differences

between real crossings and their skeletons shown in figure 1.6.

Let us now define these spaces. We will consider the associated graded spaces Afb

and Af of both flat virtual long knots and its “braid-like” variant. The braid-like variant

for virtual or flat virtual long knots is defined similarly as the usual variant except only a

subset of all Reidemeister moves are imposed. This subset consists of only the braid-like

Reidemeister moves, defined by the relative orientations of the strands and shown as R2b

and R3b as opposed to the R2c and R3c below:

R2b

R2c

R3b

R3c

Now, where as the analogous space for classical knots are chord diagrams modulo the

4T relations, the associated graded space for braid-like virtual tangles is the space of

directed chord diagrams modulo the six-term (6T) relations. A directed chord diagram,


also called an arrow diagram, is a skeleton strand with unsigned arrows ending on different

points on it, considered up to combinatorics. Two examples are shown in figure 1.7.

1

2

3

Figure 1.7. (L)An arrow diagram on a a one strand skeleton; (R) a descending

arrow diagram on a three-strand skeleton, in which all arrows point right and down.

The 6T relation is induced by the Reidemeister 3 move on the associated graded space

and is represented as follows:

6T :

+ +

-

-

-

This represents the linear combination of any six terms of arrow diagrams which differ

only at three local segments and on these are the six diagrams above. All relations below

are to be interpreted similarly.

For flat virtual long knots, the crossing flip relation induces the additional “flatness”

relation:

+=

-FLATNESS:

The associated graded spaces of the usual variant of virtual and flat virtual long

knots, in which all Reidemeister relations are imposed, have an additional XII relation:

- XII :

Finally, it can be shown these arrow diagram spaces are isomorphic to the spaces

of “descending” arrow diagrams modulo the respective “descending” relations, where


descending means that all arrows point from earlier to later points in the skeleton w.r.t.

to the orientation of the skeleton. Thus, we summarize the definitions of the associated

graded spaces Afb and Af of the braid-like and usual variants of flat virtual long knots:

_ 6T; XII:

Afb

: =

Af

: =

6T :

+

+

-

-

-

The second main results of this thesis is the following bases:

Theorem 1.0.3 (Basis of Afb1 and Af

1). A basis of Afb1 is the set of descending arrow

diagrams whose skeleton has a point before which all arrows are outgoing and after which

all arrows are incoming, as illustrated in figure 1.10. This basis is in bijection with

elements of the union⊔

n Sn of all the symmetric groups. The subspace of Afb1 of degree

k, spanned by diagrams with k arrows, has dimension k!, as verified up to degree five

in [BHLR].

A basis of Af1 is the subset of the basis of Afb

1 which excludes all diagrams containing

either illegal subdiagrams in figure 1.8. See figure 1.9 for a sample basis element and

figure 1.11 for a list for basis elements up to degree three.

Figure 1.8. Illegal subdiagrams; all arrow diagrams containing these are excluded

from the basis of Afn for any n.


Figure 1.9. Example of a diagram which is a basis element of Afb1 but not of Af

1

since the pair of chords starting from the left-most on the skeleton forms an illegal

subdiagram (see figure 1.8). Notice there is a point on the skeleton before which all

arrows are outgoing and after which are arrows are incoming

σ

... ...

if R-1 (and so XII) imposed also:if XII imposed also:

Figure 1.10. The general form of a basis element of Afb1 , Af

1 . A diagram is a basis

element of Afb1 if its skeleton has a point before which all arrows are outgoing and

after which all are incoming. It is an basis element of Af1 if furthermore it does not

contain the subdiagrams in the forbidden signs.


XII

XII XII XII XII

R1 R1

R1 R1 R1 R1R1

Figure 1.11. List of basis elements of Afb1 up to degree 3. For Af

1 , exclude the

elements with XII below them. (For the unframed version, exclude the diagrams

with R1 below them.) The chords are all directed to the right.

Finally, the above basis also generalizes to the multi-strand cases Afbn , and Af

n, which

are arrow diagrams on an n-strand skeleton modulo 6T, and modulo 6T and XII respec-

tively.

Theorem 1.0.4 (Basis of Afbn and Af

n ). A basis of Afbn is the set of descending arrow

diagrams on an n-strand skeleton in which each skeleton strand has a point before which

all arrow are outgoing and after which all arrow are incoming.

A basis of Afn is the subset of the basis of Afb

n that excludes diagrams containing the illegal

diagrams as in the one strand case in theorem 1.0.3. See figure 1.12 for a sample basis

element and figure 1.13 for a list up to degree two of basis elements in which at least one

strand is not without arrows.

Figure 1.12. A descending arrow diagram which is a basis element of both Afb2

and Af2 since it does not contain any illegal diagram. There is a point on both

skeleton strands before which all arrows are out-going and after which are all arrows

are in-coming.


XIIR1 R1 R1

Figure 1.13. List of basis elements up to degree two of Afb2 with at least one arrow

between the two strands. For Af2 , exclude the diagrams with XII below them. (For

the unframed version, exclude the diagrams with R1 below them. The chords all

point right and down.

This thesis is organized as follows. In chapter two, we introduce the general chord

diagram algebras and define the different variants of virtual and flat virtual pure tangles

in terms of it. Then, in chapter three, we give the proofs of theorems 1.0.1 and 1.0.2, the

classification of flat virtual long knots and pure tangles. In chapter four, we discuss the

notion of overlapping common multiples of chord diagrams needed in chapter six, and

in chapter five, we discuss briefly the associated graded spaces of general chord diagram

algebras and derive the the relations for the associated graded spaces of flat virtual long

knots and pure tangles. Finally in chapter six, we prove theorems 1.0.3 and 1.0.4, the

bases of the associated graded algebras.

Chapter 2

Preliminaries

In this section, we first develop the formalisms of general chord diagram algebras (−→CD)

(sec 2.1)and then define virtual and flat virtual pure tangles in terms of it (sec 2.2). We

also give the different subsets of Reidemeister moves which defines the different variants

of virtual and flat virtual pure tangles.

This definition of virtual pure tangles gives the well-known Gauss diagrams as objects,

but also an algebraic (gluing) structure among them. In chapter five, we will define

the relevant arrow diagram spaces again as general chord diagram algebras, and use

the gluing structure to in a generalized Grobner basis argument to obtain the bases in

theorems 1.0.3, and 1.0.4. However, the proof in chapter 3 of the classification of flat

virtual pure tangles (1.0.1 and 1.0.2) does not use the gluing structure, so the reader

may skip ahead to chapter 3 directly.

2.1 General Chord Diagram Algebras

2.1.1 Free General Chord Diagrams Algebra

Let a strand be the graph with one directed edge incident on two univalent vertices,

called the incoming (in) and outgoing (out) ends, according to the direction of the

13

Chapter 2. Preliminaries 14

strand. Clearly, there are two ways to glue two input strands together and output a

single strand whose orientation is well-defined and preserves those of the input, where

gluing means identifying the outgoing end of one of the input strands with the incoming

end of the other input strand and delete the identified vertex (such that the original two

edges become merged into one). Similarly, there are two binary orientation-preserving

gluing operation on disjoint union of n oriented strands with the strands ordered which

glues the ith strand of one of the inputs to the ith strand of the other input. See figure 2.1

for an example, and here is an illustration of a strand and a ordered disjoint union of

three strands:

A Single Strand Ordered-disjoint union of 3 strands

incoming

end

outgoing

end

Now, on the set of ordered disjoint unions of n oriented strands where n can be any

positive integer, we can define more ways of gluing. More precisely,

Definition 2.1.1. Let Gi be a graph of ordered disjoint unions of ni strands. For N ≥ 1,

an N -nary (orientation-preserving) gluing operation glues pairs of in and out ends of

different strands in any of the inputs G1, . . . , GN , according to some list L, provided the

strand ordering in all inputs are preserved in the output. If L is empty, this operation

will simply return an ordered disjoint union of all the strands in the input in which the

strand ordering are preserved. See figure 2.1 for an example.

Roughly then, the algebraic structure on “general chord diagrams” are these gluing

operations on the underlying skeletons of the diagrams

Definition 2.1.2 (General Chord Diagrams,−→CD). A skeleton is an ordered disjoint

union of n ≥ 1 oriented strands. A general chord diagram D on a skeleton S is a

graph consisting of the skeleton S and a finite number of another type of edges, called


INPUT 2 :

INPUT 1 :

1 1 1 2 2 2

INPUT :

INPUT 2 :

INPUT 1 :

INPUT 2 :

INPUT 1 :

1 3 1 1 2

INPUT 3 :

3 2

1 1 1 2

Concatena!on :

Binary ordered-disjoint-union :

Unary gluing :

Trinary gluing :

2

Figure 2.1. Examples of gluing operations on the set of strands: pairs of strand

ends are identified such that the orientation and the ordering of the strands in

the inputs are preserved. The numbers under a part of a skeleton defined by the

vertical divisions and strand ends denote which input it comes from. Thus, in the

concatenation operation, the three strands of input one appear in order as the three

parts labeled 1 in the output, and same with the three strands of input two. Note

that the strands in the outputs are also ordered. In the unary gluing operation, the

second and third strands of the input are glued together. In the ordered-disjoint

union operation, no strand ends are glued together but the strands of both inputs

are ordered (so that the third and fifth strands of the output are the strands in order

from input two). The trinary gluing operation is hopefully clear from the diagram.

chords, whose half edges are incident on different points on the edges of S. The chords

in D are allowed to have extra discrete data on them, for example, signs and direction.

The set of all general chord diagrams,−→CD, is the disjoint union of general chord diagrams

on all different skeletons. See figure 2.2.


-

++

+

Single-chord diagram with

decoration Ω, generators

of chord diagram algberas

Undecorated chord diagram

on 1 strand

Signed and directed chord

diagram on 4 strands

Ω

Figure 2.2. Examples of general chord diagrams.

General chord diagrams are more general than the usual chord diagrams in that the

chords can be decorated. From now on, we will simply say “chord diagrams” for general

chord diagrams. We want to consider algebraic operations on the set of chord diagrams.

These are maps from Dn1 × . . .×DnN to Dn, where Dn is the set of all chord diagrams

on a skeleton of n strands.

Definition 2.1.3 (Orientation of skeleton). Given any skeleton S, let the ordering and

the orientations of the strands be called orientation of the skeleton S.

Clearly, changing the orientation of a skeleton is such an operation on chord diagrams.

However, in this paper, we will restrict our structure to allow only operations which

preserve the orientation of the skeletons.

The most important operations on chord diagrams are the following.

Definition 2.1.4 (Gluing Operations). Let Dn be the set of all chord diagrams on

skeleton of n strands. ForN ≥ 1, anN -ary−→CD operation is a map Dn1×. . .×DnN −→ Dn

which performs an N -ary (orientation-preserving) gluing operation (as in definition 2.1.1)

on the skeletons of the inputs while preserving all chords on them. See figure 2.3.

There are also special constants in the structure, which are somewhat analogous to

identity elements.

Definition 2.1.5 (Constants). A 0-ary−→CD operation, also known as a constant, is any

empty chord diagram, a skeleton with no chords on it.


1 2

1

1 3 1 1 2 3 2

D D D3 3 3

,

D D3 2

D2 5

D3

:

,

, ,

:

: Concatena!on :

Binary ordered-disjoint-union :

Unary gluing :

Trinary gluing :

1 2 1 2

1 2 1 2 1 2

D D3

2

D2

D D5

1 3 1 1 2 3 2

1 1 1 2 2

1 2 1 2

:

Figure 2.3. Gluing operations on chord diagrams defined by the gluing operations

on strands in figure 2.1. The strands are dotted to mean that any number of chords

can end on them in any configuration, e.g. a dotted three-strand skeleton represent

any chord diagram on a three-strand skeleton. The strands in all inputs and outputs

are ordered and oriented and the operations glue together pairs of strand ends while

leaving the chords on the strands intact. Also note that only gluing operations that

preserve both the ordering and orientations of the input strands are considered. Refer

to the caption of figure 2.1 for details how the gluing is done.


Definition 2.1.6 (−→CD-Op). The set of all algebraic operations on general chord dia-

grams,−→CD-Ops, is the set of all operations generated via “composition” by the gluing

operations (definition 2.1.4) and the constants (definition 2.1.5), where composition is

defined below (definition 2.1.7). It is easy to see (and shown in section 2.1.3) that the

set of all−→CD operations are the gluing operations possibly with constants plugged in as

inputs.

Since any gluing operation which takes chord diagrams on a skeleton S as an input

can also take as an input any gluing operation or constant which outputs diagrams on

S, we can define composition of operations:

Definition 2.1.7 (Composition of−→CD-Ops). For M ≥ 0, let OM be an M -ary

−→CD op-

eration which outputs chord diagrams on a skeleton S. For N ≥ 1, let ON be an N -ary

−→CD operation whose ith input is any diagram on a skeleton S. Then OM is composable

with ON , and the composition ON i OM is the (N+M−1)-ary operation which outputs

the result of ON with its ith input as the output of OM on the ith to (i+M −1)th inputs,

and with its other inputs being the remaining N − 1 inputs in order. In particular, a

0-ary operation, or an empty skeleton, can be input to a composable N -ary operation

and the composition is an N − 1-ary operation. See figure 2.4 for an example.

Notice that the set of gluing operations (without precomposing with constants) is

closed under these compositions.


1 :

2 :

3 :

1 1 2

1' :

3 :

1 1 21' 3 3 1'

1 3 3 1 2

1 :

2 :

3 :

1 1 2

1' :

3 :

1 1 21' 3 3 1'

1 3 3 1 2

Figure 2.4. A binary gluing operation on input 1 and 2 is composed with another

binary gluing operation which now takes as its first input the output of the first

operation, renamed 1′, and as its second input input 3. The top line defines the

operations and the second line is the operations on particular inputs.

.

Finally, we define free oriented chord diagram algebras after stating some standard

terminology:

Definition 2.1.8 (Generation of chord diagrams). Given a set of chord diagrams D, the

set of all general chord diagrams−→CD generated via

−→CD-operations by D is the set of

all outputs of the−→CD-operations with diagrams in D as inputs.

Then, it is clear that given a set χ1, . . . , χn where each χi is a chord diagram

consisting of a two-strand skeleton and a decorated chord with one end on each strand

(as shown figure 2.2), called a single-chord diagram with chord decoration Ωi, the

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

chords which are decorated by any of Ωi’s.

Thus, we define the free chord diagram algebra with only order- and orientation-

preserving gluing operations:

Definition 2.1.9 (General Free Oriented Chord Diagram Algebra). The free (oriented)

chord diagram algebra−→CD⟨χ1, . . . , χn⟩ is the set of all chord diagrams generated via


all−→CD-operations (definition 2.1.6) by the set of single-chord diagrams χ1, . . . , χn along

with the−→CD-operations on them. Recall that all

−→CD-operations preserve the order and

orientations of strands in the inputs.

Remark 2.1.10. 1. The skeleton map S from−→CD-diagrams into skeletons, which for-

gets the chords in any−→CD-diagram and outputs only its underlying skeleton, com-

mutes with all−→CD-operations and in this sense is a forgetful “functor.”

2. Let the degree deg(D) ∈ Zn of a chord diagram D be the ordered set of numbers

of different types of decorated chords in D. This degree is additive under the−→CD-

operations.

2.1.2 Crossings as Chords: Another Presentation of Chord Di-

agrams

The generators of chord diagram algebras can be represented differently, and this leads

to a different set of diagrams in bijection with the chord diagrams. Namely, a single-

chord diagram can be replaced by the directed graph of a tetravalent vertex with two

incoming and two outgoing half-edges, and where each incoming half-edge is paired with

an outgoing half-edge, and these pairs of edges are ordered. The decoration of each chord

becomes the decoration of the corresponding vertex.

Then, we can define the skeleton of the tetravalent vertex graph to be the two strands

obtained from the paired edges with the vertex forgotten, and let the−→CD operations be

defined by the gluing skeleton ends as before with the output considered up to graphs.

The set of tetravalent graphs via by these−→CD-operations by these special tetravalent

vertices along with the−→CD-operations themselves are clearly isomorphic to the free chord

diagram algebra.


Ω

1

Ω

2 1 2

2.1.3 Remarks on the Chord Diagram Algebra Operations

The structure of the−→CD operations defined by the composition of operations have the

generators and axioms.

Proposition 2.1.11 (Generators of−→CD operations). The set of constants, binary ordered

disjoint union operations, and unary gluing operations which glues only one pair of strand

ends together generate (via composition) all−→CD operations due to the following.

1. Any N-ary gluing operation is the composition of an N -ary ordered disjoint union

operation which orders the strands of all N inputs with a unary gluing operation

which glues the appropriate ends together. See figure 2.5.

2. Any N -ary ordered disjoint union operation is a composition of binary ones.

3. Any unary gluing operation is a composition of unary gluing operations which glues

only one pair of strand ends together.

1 :

2 :

2 21 1

1 2 2 1

1 2 12

1 12 2

Decomposition of Binary Gluing Operation :

Figure 2.5. A binary gluing operation (the top path) is a composition of a binary

order-disjoint union with a unary gluing operation (the bottom path).


There are simple axioms satisfied by the−→CD-operations.

First, it is a structure with identity elements:

Proposition 2.1.12 (Identity Operators). For each skeleton S, the unary−→CD-operation

which outputs the input chord diagrams is an identity operator on chord diagrams on S.

These identity operators on chord diagrams remain identity elements within the structure

of−→CD operations: any

−→CD operation composed or pre-composed with composable identity

operators is unchanged.

Secondly, the operations satisfies “generalized associativity,” which roughly says that

the same picture can be drawn in any order you want, much like for concatenation,

placing an arrow on the left commutes with placing an arrow on the right. In terms

of the generating operators (as in proposition 2.1.11, the axioms are roughly that “the

generators commute.”

Proposition 2.1.13 (Axioms). There are different compositions of−→CD operations which

result in the same−→CD operation. In particular, there are

1. Unary gluing associativity as shown by example in figure 2.6.

2. Binary ordered-disjoint union associativity as shown by example in figure 2.7.

3. Unary gluing and binary ordered-disjoint union associativity as shown by example

in figure 2.8.


Unary Gluing Associativity :

Figure 2.6. Gluing different pairs of strand ends commute. The composition on

top which first glues strands 2 and 3 is equal to the composition below which first

glues strands 1 and 2.

1

3

3 1 3

2

3

2 3 3

1

2

2 1

1 :

2 :

3 :

1' :

3 :

2 1

2' :2 3 3

1 :2' 2' 1 2'

1' 3 1' 3

2 :

3 1 1

2' 1' 1' 1'

2 3 31

1' :

Binary Ordered-Disjoint Union Associativity:

Figure 2.7. Three different compositions (the top, middle, and bottom paths) of

ordered-disjoint union operations, given by the choice of which pair of input to be

ordered first, result in the same trinary operation. The arrows on the left representing

the first binary operations have their choice of two inputs labeled just before the

arrow. The intermediate output is renumbered with 1′ or 2′. If one the the inputs

are plugged in with an empty strands, then we also obtain that ordered disjoint union

with empty strands


1 :

2 :

1 2 1

2 11

12 1

Unary Gluing and Binary Ordered-Disjoint Union Associativity:

1’ :

2 :

2 1’

Figure 2.8. Gluing adjacent strands commute with ordered disjoint union provided

that the strands to be glued remains adjacent afterwards. The intermediate output

is renumbered with 1′ or 2′.

We can show now what the set of all−→CD operations including the constants are:

Proposition 2.1.14. The set of all−→CD-operations generated via composition by N-ary

(N ≥ 1) gluing operations and constants are the set of all gluing operations (as in defini-

tion 2.1.4) possibly composed with an ordered-disjoint union with finite number of empty

strands.

Proof. Using proposition 2.1.11 which gives the unary gluing operation and binary dis-

joint unions as the generators of all gluing operations, along with that any unary gluing

operation with its input fixed to be an empty skeleton (a constant in the−→CD algebra)

always returns an empty skeleton, it suffices to show that any composition of first a bi-

nary ordered-disjoint union operation in which one input fixed to be empty and then a

unary gluing operation or another binary disjoint union can be written as a “composition

in opposite order,” i.e. a path can be replaced with another in axioms 2, and 3 in the

special case with one of the inputs fixed to be an empty skeleton.

Furthermore, there are operations with left-inverses, and if we restrict the domain of

these left-inverses, then they are also right-inverses of the operations.


Proposition 2.1.15 (Inverse of Operations). 1. The set of all unary−→CD operations

which have one-sided inverses is generated by one type of unary−→CD-operations:

operations that ordered-disjoint union its input with an empty strand.

To obtain the identity operator, a unary operation of this type, which ordered-

disjoint union its input with an empty strand after the ith input strand, can be

composed with its left-inverse, a unary gluing operation which glues ith strand in its

input to either the (i − 1)th or (i + 1)th strand, at least one of which has to exist.

See figure 2.9.

2. Given a left-invertible unary−→CD operation θ as above, i.e. an operator that ordered-

disjoint unions the input with a finite number empty strands. Then, on the image

of θ, i.e. all diagrams on a skeleton consisting of at least two strands and at least

one of which is fixed to be empty, the gluing operation which is a left inverse of θ

is has θ as its left-inverse in turn. See figure 2.9.

Proof. The only part that needs proof is that the operations in ( 1) do generate all

unary operations with left inverses. By proposition 2.1.14, any unary−→CD operation

has a canonical form of the composition of first a unary gluing operation and then an

ordered-disjoint union with empty strands operation. If the first gluing operation involves

gluing together two strands both non-empty, then no unary gluing operation can undo

the gluing, the disjoint-union with empty strands operations are the only ones with left-

inverse.

2.1.4 Subdiagrams, Superdiagram, Embedded Subdiagrams

This section is needed for the second part of the paper, but put here for a more complete

discussion of the free chord diagram algebras.

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

multiplication operators Rw : A → A in any semigroup A:


Id

IdInvertible Operations :

Figure 2.9. Compositions of invertible operations. (From the left) An operation

which disjoint unions any diagram on a two-strand skeleton with an empty strand in

between the two strands is composed with a gluing operation which glues the second

strand to either the third (the top arc in the middle) or the first strand (bottom arc

in the middle) to form an identity operator. (From just before the middle two arcs)

On the image of the disjoint-union-with-empty-strand operator (the left-most short

arrow), the composition of first gluing the empty strand on either side with the

disjoint union operation which adds an empty strand in the middle of two strands,

also give the identity operator on the restricted domain.

Definition 2.1.16 (Multiplication operators). An N -nary operation in which N − 1

inputs are already fixed can be seen as an unary multiplication operator which

“multiplies” or glues the unfixed input in some fixed way with the fixed ones via the N -

nary operator. We denote such a by θa where the subscript a indexes all information of

the operator so that θa = θa′ iff θa(D) = θa′(D) for all input diagrams D. See figure 2.10.


θ

Figure 2.10. A unary multiplication operator, denoted θa : GS −→ GS′. Or

pictorially, the embedded image of d in D is considered up to sliding of its end

points on the same skeleton strand without touching any chord ends or other of its

own end points.

There are no operators which are even invertible on one diagram.

Proposition 2.1.17. The embedding of d in D determines a unique θa such that θa(d) =

D.

Definition 2.1.18 (Subdiagrams, Multiples). A subdiagram or a factor of a chord

diagram D is a chord diagram d such that there is a unary multiplication operator θa

with θa(d) = D; D is a multiple or superdiagram of d.

In semigroup theory, the same subword can be “embedded” in the same word in

different ways, e.g. the subword ab appears in aababba in two different places, a(ab)abba

and aab(ab)ba, and the multiplication operators to embed the subword into the full word

are different. In the first word, the unary multiplication operator on ab is L(a)R(abba),

where L(w) and R(w) denotes respectively left and right multiplication by the word

w. Similarly, in chord diagram algebras, we consider how a subdiagram is a factor of a

superdiagram, and the but unlike words, there is the extra point where the the unglued

diagrams are proper.

Definition 2.1.19 (Skeleton segments). Given a chord diagram D on a skeleton S, a

skeleton segment in D is an edge of the skeleton if on top of the strand ends, the chord

ends are also considered as vertices.


Definition 2.1.20 (Embedded subdiagrams). Define d as it appears in the algebraic

expression θa(d) to be the embedded image of the subdiagram d in the diagram

D = θa(d). Since D is an algebraic expression using the−→CD operations on the generators,

the single-chord diagrams, defining an embedded subdiagram of D means choosing which

single-chord diagrams inD to include, and which ends of the chosen single-chord diagrams

are glued together if they are glued together in D, and from each remaining skeleton

segment in D (between two chord ends, between a chord ends and a strand end, and

between two strand ends), how many disjoint empty strands to include. For example,

see figure 2.10.

Just like subwords can contain smaller subwords, embedded subdiagrams can contain

“smaller” embedded subdiagrams, and we can define a partial ordering on the set of

embedded subdiagrams of a given diagram D:

Definition 2.1.21 (Containment). (Recall from definition 2.1.16 θa denotes unary−→CD

gluing operator which glues some other chord diagram in some way to the input. Given

a chord diagram D, the embedded subdiagram d given by the expression D = θa(d)

contains another embedded subdiagram d′ with θa′(d′) = D if d′ is both an embedded

subdiagram of d, i.e. θb(d′) = d and its embedding in D is the composition of its

embedding in d with the embedding of d in D, i.e. θa θb = θa′. In particular, D as

embedded diagram into itself contains d.

2.1.5 Chord Diagram Subalgebras and Quotient Algebras

Definition 2.1.22. A Subalgebra of a free general chord diagram algebras is a subset

closed under all−→CD operations along with the operations.

An example is the set of all diagrams generated by some set D of diagrams, i.e. the set

of all outputs of all−→CD-operations with only the diagrams in D as inputs, or equivalently


the set of all diagrams that contains any diagram in D as a subdiagram. Notice that the

subset of all chord diagrams on the same skeleton is not a−→CD subalgebra.

Definition 2.1.23. A chord diagram algebra generated by decorated crossings

χ1, . . . , χn is the quotient of the free chord diagram algebra−→CDχ1, . . . , χn by a set

of equivalence relations which are closed under all gluing operators θa, such that the

−→CD operations descend. A chord diagram algebra presented by χ1, . . . , χn|r1, . . . , rm,

where ri is a generation relation di = d′i with di, d′i diagrams of the same skeleton, is

the quotient of the free chord diagram algebra−→CDχ1, . . . , χn by the set of all relations

θa(di) = θa(d′i), where θa is any

−→CD gluing operator. (Recall a

−→CD gluing operator glues

a fixed diagram in a fixed way to the input. See definition 2.1.16.)

Remark 2.1.24. Notice that there are sets of equivalence relations which cannot be gen-

erated by a finite set of relations of the form d = d′ but which are still closed under

all−→CD operations; for example, a subset of relations generated by d = d′ which satisfies

additional conditions at the skeleton level, say strand two of d in each relation in the

subset has to be in the same skeleton strand as strand three of d (and correspondingly

in d′. These extra conditions on the skeleton can be implemented by using only a subset

of all gluing operators to generate the relations from the equation d = d′.

Remark 2.1.25. 1. Notice that any relation that involves “smoothings” of crossings

(shown in figure 2.15) on one side of the relation is not a chord diagram algebra

relation.

2. A relation relating diagrams of the same degrees is called homogeneous, and if all

generating relations are homogeneous, then the degree of the diagram descends to

the quotient.


2.2 Virtual Pure Tangles, Flat Virtual Pure Tan-

gles,and Their Variants

We now define virtual and flat virtual long knots and pure tangles in the language of

general chord diagram algebras. This will amount to the Gauss diagram definition but

with the−→CD gluing structure on it.

Definition 2.2.1. Virtual pure tangles vPT is the chord diagram algebra

−→CD⟨−→χ +,

−→χ −,←−χ +,

←−χ −, | Reidemeister -moves⟩ generated by chords decorated with signs

and directions, shown in figure 2.12, modulo the Reidemeister relations, shown in fig-

ure 2.14. The chord diagrams for virtual pure tangles are called Gauss diagrams. See

figure ?? for an example. Virtual long knots, or virtual pure tangles on n strands,

is the subset of the vPT on a one strand or n strand skeleton respectively. Variants

of virtual pure tangles are the chord diagram algebra with the same generators as

vPT but only various subsets of the Reidemeister relations imposed. These subsets are

discussed below in section 2.2.1.

Recall from section 2.1.2 that a single-chord diagram can be represented by tetrava-

lent vertex with each incoming half-edge paired with an outgoing half edge. Now, any

directed single-chord diagram can be further represented by a planar such tetravalent

vertex in which the paired half-edges are opposite each other, and the direction of the

chord determines which of the incoming half-edge is embedded to the left of the other

incoming half-edge. If a sign is added to a directed chord, then it can be represented as

a signed planar tetravalent vertex, but we can also represent it by a crossing, as shown

in figure 2.12, where the sign represents the handedness of the crossing and the direction

denotes which strand is above the other.


+

1 2

+

1 2

1 2

2 1

-

1 2

-

1 2

1 2

2 1

Χ+:

Χ+:

Χ-:

Χ-:

PLANARGAUSSPLANARGAUSS

Figure 2.11. The four generators of virtual pure tangles represented as chords and

as crossings. The strands are ordered from left to right and the generators with “d”

below are descending.

Even though signed directed single-chord diagrams can be represented by planar

graphs, the chord diagrams generated by these may not be planar, as the−→CD gluing

operations glue the single-chord diagrams together in all possible orientation and order-

preserving ways, but not respecting planarity. Thus, when we draw these general graphs

on the plane, transverse intersections of the skeleton may appear. To distinguish these

from the real crossings corresponding to the generators of the−→CD algebra, these artifacts

are known as the virtual crossings.

-

++-

PLANAR GAUSS

Figure 2.12. The virtual knot diagram for the Kishino knot in both Gauss chord

diagram form and in tetravalent graph form immersed on the plane.

Definition 2.2.2. Flat virtual pure tangles fPT is the quotient of virtual pure

tangles vPT by the crossing-clip relation defined in figure 2.13 which equates pairs

of the generators of vPT ; and descending pure tangles dPT is the−→CD subalgebra

of vPT generated by only the descending chords (or crossings), the signed directed

chords which point from earlier to later points on the skeleton. See figure 3.2 for an


example. As in vPT , the respective long knots or pure tangles on n strands are subsets

of the respective−→CD algebra on a fixed skeleton; and variants are defined by imposing

different subsets of the Reidemeister relations.

Flat + -

1 21 2 1 2

Flat

1 2

PLANAR GAUSS

Figure 2.13. Crossing flip relation which defines flat virtual knots.

Proposition 2.2.3. The−→CD-algebra projection π : vPT → fPT splits by a

−→CD-algebra

section map that maps each of the equivalent pairs of crossings in fPT to the one which

is descending. Thus, fPT and dPT is isomorphic as−→CD-algebras.

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

to one in dPT , and it is clear that the splitting composed with the projection is the

identity.

Notice replacing the crossings in fPT always by the positive crossing is not a well-

defined section map. The same map works for the different variants of vPT /fPT /dPT

as well since the definition of the variants respects the projection map π.

2.2.1 Subsets of Reidemeister Moves

Here are the complete set of Reidemeister relations for virtual pure tangles, drawn as

Gauss diagrams and as directed tetravalent graphs with crossings as vertices.


Ɛ3

Ɛ2

Ɛ1 Ɛ3

Ɛ2

Ɛ1

Ɛ

- Ɛ

Ɛ

R1 R2 R3

R2 R3R1

PLANAR

GAUSS

Figure 2.14. The generating Reidemeister relations, where all skeleton strands can

be oriented and ordered in any way, and ϵ’s are signs corresponding to handednes of

crossings in the planar pictures.

We now define braid-like and cyclic Reidemeister-2 and 3 moves.

Definition 2.2.4. The complete orientation-preserving smoothing map is the re-

placement of all (possibly decorated) crossings in a diagram by the two skeleton strands

obtained from switching the pairing of half-edges of the crossing to the other which still

pairs an incoming with an outgoing half-edge. See figure 2.15.

Figure 2.15. An orientation preservation smoothing

Definition 2.2.5. A Reidemeister-II or III generating relation is cyclic if its image under

the complete orientation-preserving map contains a close cycle in its skeleton; otherwise,

it is braid-like. A Reidemeister relation is cyclic (resp. braid-like) if it is generated by

a cyclic (resp. braid-like) generating relation.

Note that the definitions of braid-like or cyclic are independent of the signs and

directions of the crossings/chords and so are well-defined also for the flat quotient and

descending subalgebra .


Figure 2.16. Cyclic and braid-like Reidemeister-II and III generating relations up to

the crossing flip relation

R2b

R2c

R3b

R3c

Definition 2.2.6. The braid-like variants of vPT / fPT / dPT is the quotients in

which only the cyclic R2− and R3− relations are not imposed. The framed or un-

framedversion of either the braid-like or usual variant refers to the quotient in which

R1 is excluded or included.

In the first part of this paper, we classify the framed and unframed versions of the

usual variant of flat virtual pure tangles.

Remark 2.2.7. 1. The braid-like and usual variants are indeed different, since the

cyclic R2 and cyclic R3 moves cannot be realized as a sequence of only braid-like

moves, as shown in proposition 2.2.8;

2. On the other hand, In the presence of braid-like Reidemeister moves, the cyclic R2

implies cyclic R3, since cyclic R3 is a composition of braid-like R3 and cyclic R2

moves:

R2c R3b R2c

R3c

3. There are more braid-like generatingR3 moves than cyclic ones. In the flat quotient,

there are three braid-like and one cyclic generating R3 moves.


Proposition 2.2.8. The set of all braid-like R2 and R3 relations is a proper subset of

all Reidemeister relations.

Proof. Applying the complete orientation-preserving smoothing map (in definition2.2.4)

to both sides of any braid-like R2 or R3 move will give that both sides have their incoming

open ends connected to the outgoing open ends in the same way. This is not the case for

the cyclic R2 and R3 moves.

Chapter 3

Classification of Pure Descending

Virtual Tangles

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

tangles in section 2.2, we present in this section the classification of descending virtual

pure tangles and its proof.

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

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

36

Chapter 3. Classification of Pure Descending Virtual Tangles 37

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 planar 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, we use the “thick

band” notation to represent multiple strands or arrows also shown in figure 3.1.

...

:= ...

Ɛ1 Ɛ2 Ɛk Ɛ Ɛ' ... Ɛ1' Ɛ2' Ɛm' ...

PLANAR GAUSS

Figure 3.1. An illegal interval, denoted by the skeleton interval within the square

brackets. Within the illegal interval is first a maximal under interval (in light gray)

followed by a maximal over interval (in black). Any subintervals 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.

Generically, a pure descending virtual long knot diagram has multiple maximal over

and under intervals. Due to descendingness, it always (as long as there is at least one

crossing) starts with a maximal over and ends with a maximal under interval, while in

between it alternates between over and under while having each maximal under interval

only under the maximal over intervals before it. Thus, a generic diagram has illegal

intervals on its skeleton. See figure 3.2 for an example. A generic descending virtual pure

tangle diagram is simple a descending virtual long knot diagram with finite number of

cuts on its skeleton.


+ + +

- -

-

+ +

PLANAR GAUSS

-

+

-

- + - + - - -

Figure 3.2. A generic diagram for a descending virtual long knot. The skeleton

strand can be partitioned into maximal over and under intervals. It starts with a

maximal over one, then alternates between maximal over and under, and ends in a

maximal under interval. The maximal over interval are in black, and the under in

grey in both the planar and Gauss diagrams.

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

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

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

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

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

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

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

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

Ɛ1 Ɛ2 ƐN+1 Ɛ3 ƐN

3.2 The Sorting Map

We start presenting the proof of theorems 1.0.1 and 1.0.2. Refer to page 3 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 1.2, translate

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

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

obvious.

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

-Ɛ

-Ɛ

Ɛ

Ɛ

δ δ'

δ δ'

δ δ'

δ δ'OR

F

PLANAR

GAUSS

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

Half of the R3-moves are represented by this diagram, the other half are represented

by the up-down-mirror image of this diagram

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

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

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

have become too complicated and they can be constructed easily.

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

diagram into its canonical form.

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

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

called the sorting site:

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


GF R2

Figure 3.4. (L) GF-sort; (R) R2-sort. An over (resp. under) thick band denotes

multiple over (resp. under) strands, as shown in figure 3.1 before. These sorting

moves go only in one direction.

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

sorting site.

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

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

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

Some more terminology for the definition of the sorting map.

Definition 3.2.6.

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

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

sorting move;

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

at least a crossing.

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

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

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

available sorting moves.


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

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

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

diagrams T Dvf to be

S : T Dvf −→ T Dvf

D 7−→ sk . . . s2 s1(D)

where sk . . . s2 s1 is any terminating sort sequence on D. We show below that S is

well-defined.

See section 3.5 for examples.

Proposition 3.2.8.

1. S is generated by Reidemeister-moves;

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

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

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

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

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

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

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

decreases the parameter χD by 2;

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

sorting sites.


Lemma 3.2.9. 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 1.0.2 on page 3.

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 3.2.9, divided into lemmas 3.3.1

and 3.3.2

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

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

legal intervals and the number of crossings of a diagram D ∈ T Dvf. The induction steps

will involve the diamond lemma.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 induction 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 terminat-

ing 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 relation between relations, also known as

a syzygy, st = ts, can be used; otherwise, syzygies S s = T t will be needed for the

argument.

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

R2 R2

Figure 3.5. Syzygy for overlapping R2-sorts: performing either available R2-sorts

leads to the same diagram with fewer crossings. Each sorting move happens inside

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

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

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

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

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

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

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

between them:


GF

GF

GF

GF

Figure 3.6. Syzygy for overlapping GF -sorts. Thick bands are multiple strands, as

in figure 3.1. There are two available GF-sorts to perform on the top diagram, with

their sorting sites in the vertical and horizontal boxes respectively. The path leading

from the top diagram first to the left has the GF-sort in the vertical box performed

first, followed by the only remaining GF-sort available, inside the horizontal box with

a U-shape. The path leading from the top first to the right has the GF-sort in

the horizontal box performed first, followed by its only remaining available GF-sort,

inside the vertical box. Each GF-sort lowers the number of illegal interval and both

paths lead to the same diagram at the bottom.


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

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

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

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

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

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

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

steps, with syzygies shown in figures 3.5 and 3.6. 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 3.7. Syzygy for overlapping GF and R2-sorts. Thick bands are multiple

strands, as in figure 3.1. There are two available sorts to perform on the top diagram,

a GF -sort and an R2-sort. The two paths leading from the top to the bottom

diagram corresponds to different choices of which of GF - and R2-sorts to perform

first, and result in the same diagram with fewer illegal intervals.

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


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

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

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

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

the finger-move such that they result in the same diagram, then S(Dl) = S(Dr). The

following syzygy suffices:

GF GFGF

F

Figure 3.8. Syzygy for overlapping GF -sort and finger move. Two diagrams (on

the left and right most) differing by a single F -move can be sorted by available

GF -sorts to the same diagram (at the bottom).

This completes our proof of the classification of the framed version of pure descending

virtual tangles, the first statements in theorems 1.0.1, 1.0.2.


3.4 Classification of the Unframed Version: Adding

Reidemeister I

To prove the second statements of theorems 1.0.1, 1.0.2 which classify the unframed

version 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 3.9, to the definition (3.2.7) of the sorting map.

R1

+/-

R1 R1

PLANAR GAUSS

Figure 3.9. 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 3.2.9: 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 3.10. Syzygy between R1- and R2- sorting moves.


GF

R2's

R1

R1

Figure 3.11. 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. The sorting map is applied to a generic framed descending virtual long knot diagram

below: where “deform” mean redraw the same virtual knot diagram on the plane,


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

Here the knot diagram is first “deformed” (or reimmersed on the plane) to show

the one illegal interval (in the box in the second diagram), and then GF -sorted to

remove the illegal interval, then deformed again to show the forbidden bigons, and

finally R2-sorted to remove all bigons. If the diagram represented an unframed

virtual knot, then in this case an R1-sort can be used to arrive at the canonical


form directly. Notice that the canonical form obtained this way is the same as the

one obtained by performing an R1-sort to the final diagram in the sequence above.

2. Two descending virtual pure tangle diagrams on three strands are sorted as follows:

+

+

1

2

3

1

2

3

+ +

1

2

3

+ +

+

-

GFdeform

+

1

2

3

GF-

+

1

2

3

-

+ -

deform+

1

2

3

- +

-

Note that both starting diagrams are in “braid-form,” i.e. that as Gauss diagrams,

the chords can be drawnparallel, but the canonical diagram for the first one does

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

- +

-

3.6 Remarks

1. The classification of flat virtual pure tangles can be used as an invariant on virtual

pure tangles as well as on virtual pure braids, presented by

vBn := ⟨σij |σijσikσjk = σjkσikσij , σijσkl = σklσij, 1 < i, j, k, l ≤ n ⟩


where σij can be represented by the positive crossing with strand i over strand j

as follows:

σ =i j σ =i j

-1σ =j i σ =j i

-1i

j

i

j

i

j

i

j .

The virtual braid group on n strands has an obvious map into the virtual pure

tangles on n strands.

2. We conjecture that the flat virtual pure braid group on n strand, the quotient of

the vBn by the flatness relation σij = σ−1ji , injects into flat virtual pure tangles on

n strand. If this is true, the classification above gives normal forms for the group

which are not in terms of the alphabets in the presentation of the group.

Chapter 4

Preliminaries: Minimal Common

Multiples of Chord Diagrams

In this chapter, we continue our discussion of general chord diagram algebras with the goal

of defining the two types of minimal common multiples, overlapping and non-overlapping,

of any two given diagrams. We will need to enumerate these for the leading diagrams of

the 6T, and XII relations in the proof of theorems 1.0.3, and 1.0.4 in the next chapter.

4.1 On Partial Ordering Induced by Containment

Recall the notions of embedded subdiagrams and containment in definitions 2.1.20, and

2.1.21 from page 28.

Definition 4.1.1 (Partial ordering w.r.t. containment). Given a chord diagramD, define

a partial ordering on the set of all embedded subdiagrams D by d ≥ d′ if d contains d′.

Trivial embedded subdiagrams are somewhat analogous to the identity element in a

semigroup. While constants, i.e. empty strands, are the only diagrams which can be

plugged into a binary ordered-disjoint union operation to give an invertible operator,

trivial embedded subdiagrams of d in D are essentially constants embedded in D in such

54

Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams55

a way that it can be glued to d ”in D” to give back d. Formally, followed by simply

characterization:

Definition 4.1.2 (Trivial embedded subdiagrams). Given a diagram D, and an embed-

ded subdiagram d. A constant diagram u is contained trivially in d if its embedding

in d is induced by a disjoint union with a constant operator via which another embedded

subdiagram d′ in D either contains d or is contained in d, and via whose inverse d′ is

contained in d or contains d, i.e. if u is contained in d by d = ι −→⊔(d, u) with

−→⊔(d, ·Su)

unary operation of ordered-disjoint union with d operation, so that there is d′ containing

d by d′ =−→⊔(d, u), and d containing d′ by d = ι(d′) with ι an inverse of the

−→⊔(·Sd

, u) ; or

if u is contained in d by d =−→⊔

(d′′, u), so that there is d′′ contained in d by−→⊔(·Sd′′ , u),

but also d′′ contains d by an inverse of−→⊔

(·Sd′′ , u). See figure 4.1.

Definition 4.1.3 (Open Ends of Embedded Subdiagrams). . Given a diagram D, a

skeleton segment in it (see definition 2.1.19), and an embedded subdiagram d in it. d

has an open end on the skeleton segment if it contains a a single-chord diagram one of

whose chord ends is the end vertex of the segment, or an empty strand embedded within

the segment, but does not include the entire segment, i.e. does not contain the embedded

subdiagram obtained from gluing together two single-chord diagrams whose chord ends

are the two vertices of the skeleton segment. See figure 4.1.

Proposition 4.1.4 (Characterization of trivial embedded subdiagrams). Given D and

an embedded subdiagram d. An embedded subdiagram u of D contained trivially in d is a

non-negative number of empty strands embedded in each skeleton segment of D in which

d has an open end. See figure 4.1.

Proof. The resulting diagram of gluing together two diagram can be embedded in D such

that it contains the two diagrams being glued by the gluing operation iff the strands ends

in d and d′ to be glued together are on the same skeleton segment of D.

The partial ordering is defined “up to trivial embedded subdiagrams:”


d':=

u:=

d:=

Figure 4.1. d, u and d′ are the subdiagrams embedded inside the square brackets

of the same overall superdiagram. d (or respectively d′ and u) has an an open end

on each skeleton segment of the overall diagram which has at least a half square

bracket on it. u is contained trivially in both d (resp. d′), since it consists of empty

strands embedded only in the skeleton segments of the superdiagram in which d

(resp. d′) has open ends. d and d′ are equal in the partial ordering by subdiagram

containment since they contain the same chord and have open ends on the same

skeleton segment in the overall diagram.


Proposition 4.1.5 (Embedded subdiagrams equal under partial ordering by the con-

tainment). Given a diagram D and an embedded subdiagram d. An embedded subdiagram

d′ contained in d will also contain d, i.e. d′ equal d w.r.t. to the p.o. by containment, iff

if d′ contains the same single-chord diagrams and have open ends on the same skeleton

segments, i.e. if it only differs from d by containing different number of empty strands

from each skeleton segment in D on which d has an open end, on the condition that it

still includes all disjoint empty skeleton segments of D that d contains. See figure 4.1.

Proof. If d contains d′ and d′ contains d, then there are gluing operators θa and θa′ with

θa′(d′) = d and θa = d′, meaning θa θa′(d′) = d′, and these have to be the invertible

operators, which are generated by composition by the ordered-disjoint union with empty

strands operators and glue empty strands on one end to adjacent strands operators as in

proposition 2.1.15. The rest is by the definition of trivial embedded subdiagram.

In contrast:

Proposition/Definition 4.1.6 (proper embedded subdiagrams). Given D, an embed-

ded diagram d embedded in it, and another embedded diagram d′ contained in d. d′ is

contained properly in d whenever it is strictly smaller that d w.r.t. partial ordering by

containment. d′ is proper in d iff it

1. includes strictly fewer single chords of D than d;

2. or otherwise does not glue together strand ends of single-chord diagrams that is

glued together in d;

3. or otherwise excludes a skeleton segment of D which d does not exclude.

In particular, a proper embedded subdiagram d of a diagram D is one which is

properly contained in D as an embedded subdiagram of itself. If D has no empty strands,

then d either does not include one of the chords in D or does not glue together the strand

ends of two single-chord diagrams which are glued together in D. See figure 4.2.


Remark 4.1.7. The partial ordering is preserved under multiplication of D by other dia-

grams, i.e. for d,d′ embedded subdiagrams of D, if d ≤ d′ in D then d ≤ d′ as embedded

diagrams in the bigger diagram θa(D) for any multiplication operator θa.

On the other hand, there can be new order relations between embedded subdiagrams

of D in superdiagrams of D; in particular that properness may not be preserved. But this

only happens when an embedded subdiagram d′ is strictly smaller than d only because d′

excludes one or more skeleton segments of D which d does not exclude and the excluded

skeleton segment is adjacent to one that is not excluded by d′, then and only then in any

superdiagram of D in which all the excluded segments are glued to the adjacent segments

which are not excluded, d′ contains d as well. In particular, properness is not preserved

by multiplication.

D:=

d := 1

d := 2

d := 3

Figure 4.2. d1, d2, d3 are embedded subdiagrams of D where D > d1 > d2 > d3.

d1 does not contain the last disjoint skeleton segment of D; d2 does not include the

“first” chord in d1; and d3 does not glue together the two chords which are glued

together in d2.


4.2 Common Multiples, and Common Factors

Recall from section 13 (page 13) that θa : D(n) −→ Dm is a unary gluing operator on D

which glues the input with some fixed diagram in some fixed way, which the subscript

a indexes, that a diagram d is a subdiagram of a diagram D if there is θa such that

θa(d) = D (see figure 2.10).

Now, we define common multiples (CM), minimal common multiples (MCM), and

MCM’s with non-trivial common factors, equivalently overlapping MCM’s, notions anal-

ogous to those in semigroup theory.

Definition 4.2.1 (Common multiple (CM), factorized CM). A common multiple of

diagrams d and d′ is a diagram L which contains both d and d′ as embedded subdiagrams,

i.e. L = θa(d) = θa′(d′) for some unary multiplication operators θa and θa′ . We will call

the diagram L along with the embedding information given by the expressions L =

θa(d) = θa′(d′) a factorized common multiple. See figure 4.3.

Definition 4.2.2 (Embedding of factorized multiples of the same diagram). embedding

of a factorized multiple into another factorized multiple. Given two factorized multiples

M = θa(d) and m = θa′(d) of the same diagram d. An embedding of the factorized

common multiple m into the factorized common multiple M is an embedding θb

of m as a diagram into M such that the embedded image of d given by the embedding

of m matches the embedded image of d in M given by θa(d) = M , i.e. θb θa′ = θa. See

figure 4.3

Minimal common multiples roughly look like “unions” of the factor diagrams.

Definition 4.2.3 (MCM’s, factorized MCM’s). Given two chord diagrams d and d′.

A factorized common multiple L = θa(d) = θa′(d′) is a minimal common multiple

(MCM) of d and d′ if it is minimal (w.r.t. containment) among all of its embedded

subdiagrams which also contain both d and d′, i.e. any proper embedded subdiagram of L


d:= d':=

( ) ( ) ( )Overlapping CM(d, d’):

Overlapping MCM(d, d’):( ) ( ) ( )

( ) ( () )

Non-Overlapping CM(d, d’):

Non-Overlapping MCM(d, d’):

( ) ( ) ( )

( ) ( ) ( )

Figure 4.3. Examples of common multiples and minimal common multiples of the

same diagram d and d′. The square and round brackets denote the embedding of

d and d′ respectively. The first CM is not minimal since it contains a chord and

a disjoint skeleton strand not belonging to both d and d′, while the second CM is

the unique MCM contained in the first one. The overlap diagram in the first two

common multiples is the single-chord diagram with the dotted dark chord. The third

CM is not minimal since it glues together strand ends which are not glued together

in either d or d′ already, but the fourth CM is the unique MCM contained in the

third. There is no non-trivial overlap diagram in these last two CMs.


cannot contain both d and d′. L = θa(d) = θa′(d′) along with the embedding information

is called a factorized minimal common multiple of d and d′. See figure 4.3.

We will use chord to mean the embedded single-chord diagram of which the chord is

an edge.

Proposition 4.2.4 (Characterization of factorized MCM). A factorized common mul-

tiple of d and d′ is minimal iff

1. each of its chords is contained in at least one of d or d′;

2. any pair of skeleton ends of its single-chord diagrams is glued together only if is

glued together in either d or d′;

3. each of its disjoint empty strands contains at least one empty strands of either d or

d′.

See figure 4.3.

Proof. Apply proposition/definition 4.1.6.

Remark 4.2.5. 1. If L is a factorized MCM of d and d′, it is also minimal among all

embedded subdiagrams in any superdiagram θa(L) which contains the embedded

images of d and d′. Conversely, if any embedded subdiagram is a factorized common

multiple of d and d′ is minimal embedded factorized common multiple, then it is

equal w.r.t to partial ordering by containment to an embedded factorized common

multiple that is an MCM.

2. The set of all MCMs of two diagrams d and d′ is finite because all combinatorics

are finite.

We will now restrict our attention mostly to d and d′ which have no disjoint empty

starnds for simplity.


Proposition 4.2.6 (Generating set of CMs). Given two diagrams d and d′, the set of

all factorized MCM’s generate all factorized common multiples of d and d. If d and d

both do not have disjoint empty strands, each factorized common multiple L = θa(d) =

θa(d′) contains a unique minimal factorized common multiple, the factorized embedded

subdiagram which:

1. includes all single-chord diagrams contained in either d or d′

2. glues the ends of two single-chord diagrams together if they are glued in either d or

d′ (but does not glue if not glued in neither).

It is clear that the set of all MCM’s of d and d′ is finite.

Proof. Apply the characterization in proposition 4.2.4.

If d or d′ has disjoint empty strands, then a factorized common multiple of d and d′

can contain different factorized minimal common multiples of d and d′, but we will not

be concerned at the moment.

We distinguish common multiples L(G,θa),(G′,θa′ ) into two types, the “overlapping”

and “non-overlapping. ” This is analogous in semigroup to words acd is a common

multiple of ac and cd which has an “overlap,” (or common factor) c, but the word abcd

is a common multiple of ab and cd which has no overlap or only a trivial one of e does

not overlap. For words, we can only distinguish between overlapping on the left, right,

or middle, but evidently for chord diagrams, there are many more choices for “where”

the overlap is. The overlap diagram is essentially the “intersection” of the two embedded

subdiagrams defined compatibly with the algebraic structure of chord diagrams.

Definition 4.2.7 ((Maximal) Common embedded subdiagram). Given chord diagrams d

and d′, and a factorized common multiple L = θa(d) = θa′(d′). A common embedded

subdiagram of d and d′ in L is any embedded subdiagram of L contained in both


d and d′. A maximal common embedded subdiagram of d and d′ in L is a

common embedded subdiagram which is maximal w.r.t. containment among all common

embedded subdiagrams of d and d′ in L.

Maximal common embedded subdiagrams of any given factorized common multiple

are not unique and can differ by trivial embedded subdiagrams as in proposition 4.1.5, but

if one of these maximal common embedded subdiagrams are trivial in some superdiagram

then all others also are.

Definition 4.2.8 (Overlap diagram, Overlapping factorized MCM). Two diagrams d and

d′ overlaps in L = θa(d) = θa′(d′) if all the maximal common embedded subdiagrams of

d and d′ are not trivial (as in definition 4.1.3) in at least one of d and d′. In this case, we

call the maximal common embedded subdiagram with the fewest disjoint empty strands

the overlap diagram of L, and L an overlapping factorized M.C.M. See figure 4.3.

Proposition 4.2.9 (Characterization of overlap diagram). An embedded diagram of a

factorized MCM L = θa(d) = θa′(d′) of two diagrams d and d′ is the overlap diagram of

diagram of L iff it :

1. contains all chords contained in both d and d′;

2. glues strand ends of single chord diagrams in L which are glued together in both d

and d′;

3. includes an empty strand in all disjoint segments in L contained in either d or d′.

See figure 4.3.

Chapter 5

Preliminaries: the Associated

Graded Spaces

In this section, we give a brief introduction to the universal finite-type invariant theory of

general chord diagram algebras following Bar-Natan as a motivation to study the spaces

in theorems 1.0.3 and 1.0.4 (sec 5.3). We explain briefly the derivation of Av and Af and

define the infinitesimal algebras Av and Af of virtual and flat virtual pure tangles as a

general chord diagram algebra (sec 5.6). The algebraic structure of these general chord

diagram algebras are used in the proofs of theorems 1.0.3 and 1.0.4 (sec 6.1).

5.1 Graded Spaces Associated to Filtered Vector Spaces

Here are some standard notions for any filtered vector spaces.

Definition 5.1.1. Given a filtered vector space V = I0 ⊇ I1 ⊇ I2 . . .. The completed

associated graded vector space associated to V is

Gr V := I0/I1 ⊗ I1/I2 ⊗ . . . .

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

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

64

Chapter 5. Preliminaries: the Associated Graded Spaces 65

Proposition 5.1.2.

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

Gr Z : Gr V → Gr V = Id.

Gr Z does not depend on Z |I∞ where I∞ :=∩

n In; V/In is isomorphic to I0/I1 ⊕

. . . In−1/In; and Z is in general neither surjective nor injective.

Proof. We construct a map Z. Let πi : V → V/Ii be the projections. Choose a sequence

of linear section maps γi : Ii/Ii+1 → Ii | i ∈ N, γi(Ii+1) = 0 (Notice there is no

canonical choice for this.) Define

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

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

5.2 Linear Extension of Chord Diagram Algebras

Any chord diagram algebra can be extended linearly over a field K (say of characteristic

0) in slightly more complicated way as a semigroup is extended linearly to an associative

algebra. We take the linear combinations of all diagrams Dn on the same skeleton and

then take the union of these. We do not allow linear combinations of diagrams on different

skeletons.

Definition 5.2.1 (Linear Extended Algebra). The set of objects in a linearly-extended

free−→CD-algebra is the union ⊔nKDn over n of the K-vector spaces spanned by dia-

grams on the same n-strand skeleton. In the linearly-extended free−→CD-algebra, the

−→CD-operation from KDn1 × . . . × KDnN

to KDnMis lifted to N -linear maps on tensor

products KDn1⊗ . . .⊗KDnNto KDnM

; and consequently, any unary−→CD-gluing operator

θa, which glues a fixed chord diagram in a fixed way to the inputs (see figure 2.10), is

also lifted to a linear map KDnNto KDnM

.


Definition 5.2.2 (Ideal). An ideal in the linearly extended−→CD algebra is a union ⊔nvn

of subspaces vn ⊆ KDn which is closed under all linearly-extended unary multiplication

operator θa. The set of equivalence relation generated by any equation d = d′ with

d, d′ ∈ Dn is lifted to the union of subspaces spanned by θa(d − d′) for θa a linearly-

extended unary gluing operator.

As in associative algebras, the quotient of the linearly extended−→CD-algebra by an ideal

retains the algebraic operations from the free−→CD-algebra. The presentationK

−→CD⟨χ1, . . . , χn |

r1, . . . , rm⟩ where χi is a single-chord diagram with possible decoration and ri is a linear

combination of diagrams of the same skeleton, to mean the linear extended free chord

diagram algebra modulo the ideal generated by r1, . . . , rm.

5.3 Brief Introduction to Finite Type Invariant The-

ory

In this section, we introduce the theory of finite-type invariants on chord-diagram al-

gebras following Bar-Natan and define the arrow diagram algebras Af, and Afb as the

target spaces of universal finite-type invariants of the different variants of flat virtual pure

tangles. The theory generalizes notions in associative algebras, but can be generalized to

much more general algebraic structures, but we will focus on the theory for−→CD-algebras.

From now on we may abuse notation and call any linearly-extended−→CD-algebra sim-

ply by−→CD-algebra, and similar omit the phrase “linearly-extended” for its operations.

In particular, we will use vPT and fPT for the linear extended−→CD-algebras of virtual

and flat virtual pure tangles respectively.

Let T be any−→CD-algebra, KT be its linear extension, and I any ideal in T .


Proposition/Definition 5.3.1. 1. Let I be an ideal in KT . For n > 1, define the

nth power of I to be the union of subspace spanned by all outputs of operations

with at least n inputs in I, and denote it by 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

−→CD operations are filtered, i.e. the output of an operation with inputs respectively

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

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 a new operation with

the same N inputs in I.

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

Proposition 5.3.2. Let Gr I KT be the associated graded space w.r.t. the filtration by

powers of an ideal I. Any N -nary−→CD-operation on KT induces an N-nary operation

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

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

i=1 ki, and satisfy the same

axioms, such as generalized associativity, as the−→CD operations.

Proof. Any operation induced by a−→CD operation will output an element in

Ik1+...+kN/Ik1+...+kN+1 when the inputs are from Ik1/Ik1+1, . . . , IkN/IkN+1 respectively.

The induced operation is well-defined since the−→CD 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 .

The induced operations satisfy all the axioms of the−→CD operations since they have the

property that these on the diagrams imply the axioms with linear combinations of the

diagrams as inputs, and the generators of any ideal has to be linear combinations of

diagrams.


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

and the associated graded spaces where the filtration is given by a specific canonical

ideal.

Proposition/Definition 5.3.3 (Augmentation ideal). In any−→CD-algebras, the union

of subspaces spanned by the formal differences d− d′ of diagrams d and d′ on the same

skeleton is an ideal, called the augmentation ideal.

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

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

difference of two objects of the same kind.

Definition 5.3.4 (Expansion). Let I be the augmentation ideal. We call a union of

linear maps Z : KT → Gr IKT as in proposition 5.1.2 an expansion, or a universal

finite-type invariant. A homomorphic expansion is an expansion that respects all

−→CD operations.

5.4 Associated Graded Spaces of Free Chord Dia-

gram algebras

We now find a presentation of the associated graded space of any free−→CD algebra FT :=

K−→CD⟨χ1, . . . , χn⟩. These results are in [GPV]. Again, “operators” will mean the linearly-

extended ones.

First, the generators of the augmentation ideal.

Proposition 5.4.1. The augmentation ideal IF of any linearly-extended free chord dia-

gram algebra, K−→CD⟨χ1, . . . , χn⟩ is generated via all unary gluing operators θa by the formal

differences χi := χi − S(χi) of the generators with their skeletons, called semi-virtual

chords, which are drawn dashed. See figure 5.1 for an example.


Proof. Any difference of diagrams on the same skeleton can be written as a telescopic

summation of differences of diagrams which differ by only one 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.

Since each generator χ can be written in terms of the semi-virtual chord χi, we may

use another “basis” for the free chord diagram algebra which may be more convenient

for the analysis of its associated graded w.r.t. the augmentation ideal. Here basis really

means a union of sets each of which is a basis for each of the vector spaces in the free

chord diagram algebra.

Proposition 5.4.2. The set of all diagrams generated (via−→CD operations) by χii forms

a vector space basis for the free chord diagram FT .

Proof. These diagrams clearly span since any diagrams in the free chord diagram 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 the original

basis of diagrams in the original generators χ. 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 one 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.

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

Proposition 5.4.3. 1. For all n ≥ 0, the nth power of the augmentation ideal InF

has basis the set of all diagrams with n or more semi-virtual chords. Thus, the

quotient InF/In+1F for all n has a basis in bijection with all diagrams with exactly n

semi-virtual crossings/chords.


2. The graded space GrIFFT associated to the filtration of the FT by powers of the

augmentation ideal IF is the free−→CD algebra K

−→CD⟨χ1, . . . , χn⟩ generated by the

semi-virtual chords χi.

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

homomorphic expansion.

Proof. Simple checks.

Remark 5.4.4. This is a generalization of the case of a free finitely-generated associative

algebra K⟨x1, . . . , xn⟩. The augmentation ideal IF is generated by the formal differences

xi := xi− 1 of the generators with the identity element, and the associated graded space

w.r.t the filtration by powers of IF is the the free finitely-generated associated algebra

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

5.5 The Associated Graded Spaces of Chord Dia-

gram Algebra

We now turn to the question of determining the associated graded space of a quotient of

free chord-diagram algebra.

Proposition 5.5.1. Let T := K−→CD⟨χ1, . . . , χn | r1, . . . , rk⟩ be the quotient of the free

−→CD-

algebra FT := K−→CD⟨χ1, . . . , χn⟩ by the ideal R generated by the relations r1, . . . , rk. Let

IF and I be the augmentation ideals of FT and T respectively. Then GrT ∼= GrFT /R

(as−→CD-algebras) where R := ⊔nRn with Rn := (R ∩ InF + In+1

F )/In+1F for all n. Thus,

to determine GrT is equivalent to determining Rn for all n.

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

F +R)

= InF/((In+1F +R) ∩ InF ) = (InF/In+1

F )/((R∩ InF + In+1F )/In+1

F ).

Proposition 5.5.2 (Relations for GrT ). 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 /InF , then ri + InF is one of the generating relations of GrT . In general, these may

not generate all of R.

Proof. From definitions.

5.6 The Associated Graded Spaces of vPT , fPT , and

dPT

We will look for relations in the associated graded spaces Afb and Af of the framed

versions of the braid-like and usual variants of flat virtual pure tangles. They will turn

out to be the defining relations, but we will only prove this in a later paper.

Recall from page 30 that virtual pure tangles

vPT :=−→CD⟨−→χ +,

−→χ −,←−χ +,

←−χ −, | Reidemeister-moves⟩ is the quotient of the free−→CD

generated by chords decorated with both signs and directions modulo the Reidemeister

relation, and flat virtual pure tangles fPT is a quotient of vPT by the crossing-flip

relations. The framed version of the braid-like variant means that only the braid-like R2

and R3 relations are imposed, while the framed version of the usual variant means that

all R2 and R3 relations are imposed.


Now proposition 5.4.1 gives that Gr vPT and Gr fPT are generated (via the−→CD

gluing operations) by the four semi-virtual crossings each corresponding to a generator

of vPT as shown in figure 5.1.

:= :=_ _

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

PLANAR

+_

:= +_

:=_ _

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

GAUSS

Figure 5.1. Semi-virtual chords crossings, generators for Gr vPT and Gr fPT .

We now follow proposition 5.5.2 (which was also done in [GPV]), and project the R2

and R3 relations and flatness relations to the lowest degree by first rewriting the crossings

in terms of the semi-virtual crossings by

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

Under application of proposition 5.5.2, the Reidemeister 2 moves, both braid-like and

cyclic, give the following relation between pairs of semivirtual crossings:

+=

--

From now on, we will use this relation to eliminate the negative semi-virtual crossings

from other relations. Applying proposition 5.5.2 to any of the Reidemeister 3 moves, we

obtain the same eight-term relation:

=+ ++ + + +

of which the lowest degree gives six-term (6T) relation:


6T :

+ +

-

-

-

For fPT , the extra flatness relation gives the following flatness relation between the

two positive semi-virtual crossings with different directions:

+=

-FLATNESS:

Finally, following [BHLR], there is a generating relation in GrvPT and GrfPT not

prescribed by proposition 5.5.2 using our presentation of vPT . This is the XII relation

in figure ?? induced from the difference of a braid-like and a cyclic R2 move:

Summarizing, and leaving the proof that the above relations indeed generate all re-

lations in the associated graded spaces, we have

Gr vPT b ∼= Avb :=−→CD⟨−→χ ,←−χ , | 6T ⟩

Gr vPT ∼= Av := Avb/⟨XII⟩

Gr fPT b ∼= Afb := Avb/⟨Flatness⟩

Gr fPT ∼= Af := Avb/⟨XII,Flatness⟩

(5.1)

In fact, the projection from Av(b) −→ Af(b) by the flatness relation splits:

Proposition 5.6.1. The projection from Av(b) onto Af(b) has a right inverse given by the

section map that maps the equivalent pair of semi-virtual chords/crossings −→χ and ←−χ to

the one which is descending, i.e. pointing from an earlier to a later point in the skeleton.


The section map is an−→CD algebra map and so Af(b) is isomorphic to the

−→CD subalgebra

in Av(b) generated by the descending semi-virtual chord.

Proof. It suffices to check that any 6T and XII relation is mapped to a relation in

Finally, the following are the−→CD-algebras Avb and Av which are the associated graded

spaces of the framed version of fPT b and fPT respectively, whose bases are the second

main results of this paper with proofs in section 6.3:

1 2 n

...

3

+ + 6T:

1 2 n

...

3

+ + _

6T: XII:,

Avfb

: =

Avf

: =

Figure 5.2. Summary of definitions of Afb and Af. The strands are in descending

order from left to right in the 6T relations.

Chapter 6

Proof of Bases of Afb and Af

In this chapter, we prove the second main results of this paper, theorems 1.0.3, 1.0.4 on

page 9, which give bases for the chord diagram algebras Afb, Af, which are the associated

graded spaces of the framed version of the braid-like and usual variants of flat virtual

pure tangles (as explained in section 5.6). We will first describe a general argument in

section 6.1 and then apply this general argument to our specific−→CD algebras.

6.1 Grobner Argument for Chord Diagram Algebras

This section provides an answer to special cases of the following general question: What

is a vector space basis for each of the spaces in a chord diagram algebra presented by

A :=−→CD⟨χ1, . . . , χn |r1, . . . , rm ⟩? Our main observation is that the Grobner basis argu-

ment for associative algebras can be adapted to chord diagram algebras essentially by

replacing the role of associative multiplication by the unary CTD-multiplication oper-

ators θa (as in definition 2.1.16) It turns out that our−→CD-algebras of interest, Afb and

Avf, admit such generalized Grobner bases.

Our main statement is lemma 6.1.4, which follows after a restatement of infinite di-

mensional Gaussian elimination and a lemma on the generators of differences of spanning

75

Chapter 6. Proof of Bases of Afb and Af 76

vectors of an ideal I in a general chord diagram algebra.

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 6.1.1 (Infinite Dimensional Gaussian Elimination). Given a vector space V of

all finite linear combinations of elements in a countable basis B, and a subspace W which

is the span of a (possibly infinite) set SW := w1, w2, . . ., if there is a partial ordering

on B such that w.r.t. it

1. any descending chain in B is finite;

2. each vector w in the spanning set SW has a unique leading term, denoted aw lw

where aw ∈ K and lw ∈ B;

3. the difference of each pair of vectors wi, wj in the spanning set SW which have

proportional leading terms, i.e. cawilwi

= awjlwj

where c ∈ K, can be written in

terms of vectors in SW with strictly lower leading terms as follows:

(cawiwi − awj

wj) =m∑k=1

awkwk where lwk

< lwi∀k

and call the set of all leading basis elements L := lwi∈ B | wi ∈ SW.

Then any subset of SW consisting of vector representatives of the leading basis elements,

i.e. for each l ∈ L, choose one wi ∈ SW with lwi= l, is a basis of W, and B − L is a

basis of V/W.

Proof. This is elementary linear algebra. Use induction on the partial ordering for both

statements. Condition 1 and that each vector in V consists of only finite linear combina-

tions ensure all needed algorithm terminate.


Like the usual Grobner basis argument on (non-commutative) associative algebras,

the Grobner basis argument for−→CD below improves the above, which demands infinitely

many syzygies, by using the property of the algebraic structure that “common multiples”

of any two elements are generated by “minimal common multiples” to reduce the search

for syzygies at the generator rather than vector level.

But first, given A :=−→CD⟨χ1, . . . , χn |r1, . . . , rm ⟩, we will want to find all pairs of

relations θa(r), θa′(r′) with the same leading diagram.

Definition 6.1.2 (Overlap type of pairs of relations). LetA :=−→CD⟨χ1, . . . , χn |r1, . . . , rm ⟩.

Suppose that for each relation ri there is a special diagram among all terms in the ri

called the leading diagram Li, i.e. ri = aLi +∑

ckDk where a, ck are coefficients and Li

and Dk are diagrams on the same skeleton. Then for each pair ri, rj of relations (i = j

possibly), and each overlapping factorized MCM (refer to definition 4.2.8) of the leading

diagrams of the two relation, namely the algebraic expression L = θa(Li) = θa′(Lj), we

call the pair of expressions θa(ri) = θa′(rj) of relations generated by ri and rj with the

same leading diagram an overlap type of the generating relations. We also denote

the difference up to scalar multiple of (cθa(ri) − θa′(rj)) where c ∈ K is such that the

leading terms of cθa(ri) and θa′(rj) cancel by δ(ri,θa),(rj ,θa′ ), the leading diagram in the

cancelling terms L(ri,θa),(rj ,θa′ ).

Definition 6.1.3. A partial ordering on the diagrams G is said to respect the−→CD structure

if for any G,G′ ∈ G

G ≤ G′ ⇐⇒ θa(G) ≤ θa(G′) ∀−→CD operations θa

Lemma 6.1.4 (Grobner Bases for Chord Diagram Algebras). Given a (linearly-extended)

−→CD-algebra A :=

−→CD⟨χ1, . . . , χm |r1, . . . , rn ⟩ = D/I, where D is the set of all diagrams

in A and I the ideal generated by the relations r1, . . . rn. (Recall that each relation ri

is linear combination of diagrams on the same skeleton.) Then if there exists a partial


ordering O on the set of all chord diagrams which respects the−→CD-structure, and such

that w.r.t it:

1. any descending chain in D is finite;

2. each generating relation ri has a well-defined unique leading term; and let the set

of all leading diagrams be L;

3. for each pair of generating relations ri, rj (i = j possibly), and each overlap type

θb(ri), θb′(rj) of ri, rj, there exists a generating syzygy

δ(ri,θb),(rj ,θb′ ) =m∑k=1

ck θak(rk) where ck ∈ K (6.1)

such that the leading diagram L(ri,θb),(rj ,θb′ ) which will be canceled on the L.H.S. is

a (well-defined) unique maximum among the leading diagrams of all the relations

θak(rk) on the R.H.S. in the syzygy.

Then for each skeleton S, a basis of each vector subspace IS := I∩DS is any subset of

relations in it which has a bijection with the set of leading diagrams LS := L∩DS , where

L was given in condition 2, and a basis for each space A∩DS of different skeleton in

the chord diagram algebra A is the set of all chord diagrams on skeleton S not in LS.

Proof. We apply the lemma 6.1.1 with V = GS,W = IS and the spanning set SIS ofW to

be the set θa(gi)∪GS of all multiples of the generating relations g1, . . . , gn with skeleton

S. It suffices to show that for any two relations r = θa(gi), r′ = θb(gj) in the spanning

set SIS with the same leading diagrams, there exists a syzygy such that the difference

cr − r′ can be written as a linear combination of relations in S all with lower leading

terms (w.r.t. O). Now, since the partial ordering on G respects any−→CD-multiplication

operation θa, it suffices to show there are such syzygies for a set that generates all pairs

of relations with same leading diagrams, and we know a finite generating set from the

enumeration of the factorized MCM of pairs of leading diagrams of the relations.


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.

6.2 Partial Orderings on Chord Diagrams

Before applying the above argument to our specific general chord diagram algebras, let

us describe a general way of defining partial orderings on G.

First, counting the number of ways a given general chord diagram algebra g ∈ G can

be embedded in any general chord diagram algebra in G gives an ordering on G:

Definition 6.2.1. Let G1, G2, . . . Gk ∈ G be general chord diagram algebras 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 6.2.2. 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 6.1.4, 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 general chord diagram algebras in G (i.e. those needed in lemma 6.1.4), 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 6.1.4 as conditions

on (NG1 , NG2 . . . NGk):

Lemma 6.2.3. A partial ordering O on G which satisfies all conditions required in

lemma 6.1.4 can be constructed from the ordering induced by an ordered set of functions

(NG1 , NG2 . . . NGk) if the set satisfies the following:

1. for each gj =∑

s asHs ∈ R, there is one special diagram Lgj = Hs among all

diagrams Hs such that for some 1 ≤ n ≤ k and for all θa,

∀ i < n NGi(θa(Lgj)) = NGi

(θa(D)) and NGn(θa(Lgj)) > NGn(θa(D))

where D is any of the diagrams Hs = Lgj ;

2. for at least one syzygy (assuming existence) for each overlap type (gi, θa), (gj, θb)

of any pair of generating relations gi, gj, there is some 1 ≤ n ≤ k such that for all

θa,


∀ i < n, NGi(θa(L(gi,θa),(gj ,θb))) = NGi

(θa(Lr)) and

NGn(θa(L(gi,θa),(gj ,θb))) > NGn(θa(Lr))

where Lr is any of the leading diagrams other than L(gi,θa),(gj ,θb) appearing in the

syzygy.

Proof. All follows directly from definitions.

Furthermore, we can reduce the check of whether the partial ordering induced by a

counting function NG is preserved under unary gluing operators to the finite conditions

in the next lemma 6.2.5

Definition 6.2.4. Let NG be a function that counts the number of embeddings of G in

a diagram (as in definition 6.2.1). Given a diagram D and a fix unary gluing operator

θa, let NG(θa(D)) be the number of different embeddings θa′ of G into the θa(D) such

that the factorized common multiple θa′(G) = θa(D) is minimal and overlapping, i.e.

NG(θa(D)) := |θa′ | θa′(G) = θa(D)| is a factorized overlapping MCM of G and D |

(Clearly, if there is no overlapping MCM between D and G which contained D by θa,

then NG(θa(D)) = 0.)

Lemma 6.2.5 (Sufficient Conditions for Ordering Respected by−→CD-Multiplication).

Given two diagrams L and D with the same skeleton. To show that D ≤ L under the

partial ordering induced by NG and that this ordering is preserved under gluing with other

diagrams, i.e. NG(θa(D)) ≤ NG(θa(L)) for all unary gluing operators θa, it suffices to

show that NG(θa(D)) ≤ NG(θa(L)) only for each of those θa’s such that NG(θa(D)) > 0,

i.e. there exists an overlapping factorized MCM of G and D which contains D by θa.

If moreover NG(Id(D)) < NG(Id(L)), then NG(θa(D)) < NG(θa(L)) for all unary

gluing operators θa.


On the other hand, for NG(θa(D)) = NG(θa(L)) for all unary gluing operators θa, it

suffices that NG(θa(L)) = 0 =⇒ NG(θa(D)) = 0 for any θa, i.e. for any θa such that

there is an overlapping factorized MCM of G and L containing L by θa, there also exists

overlapping factorized MCM of G and D containing D by θa, and for each θa such that

NG(θa(D)) > 0, NG(θa(D)) = NG(θa(L)). See figure 6.1 for an example.

Proof. All embedding of G into θa(D) gives a common multiple which contains a unique

factorized MCM of G and D, so we can count the embeddings in groups with the same

factorized MCM.

For any given 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.

We further count these in groups that give the same factorization θa in D but maybe

different factorization of G. This is because given that L and D have the same skeleton,

for each factorization θb in D, the number of embeddings of θb(D) into θa(D) is just

equal to the number of embeddings of θb(L) into θa(L), both being | θa′ θb = θa |,

and the number of the embeddings of G in the given factorized multiple θa(D) is the

sum∑

θb| θa′ | θa′ θb = θa | × NG(θb(D)) where the sum is over all gluing operators

θb such that θb(D) is an overlapping MCM of D and G which contains D embedded by

θb. The inequalities/equalities between NG(θa(L)) and NG(θa(D)) for all θa’s then follow

from simple enumeration.


L:= D:= HT:=

θ :=

θ (L)= θ (D)=

Gluing Operators θ for which θ (L) or

θ (D) is an overlapping MCM with HT

a

a a

b

1 chord

overlap

No overlap

2 chord

overlap

b

θ : b

θ : b

θ : b

b

θ : b = Id

Embeddings of HT s.t. θ (L)/θ (D) is an MCM which

contains L/D by θ

G L / D

G G

b

θ (L) θ (D) b b

G G

G G

G

L

L

L

L

D

D

D

None

b

b

Gluing Operators θ for which θ (L) or

θ (D) is an overlapping MCM with HT

b

1 chord

overlap

No overlap

2 chord

overlap

b

θ : b

θ : b

θ : b

b

θ : b = Id

θ a=θ a’ θ bθ s.t. a’

Figure 6.1. List of different types of multiplication operators θb


6.3 Restatement of Bases for Arrow Diagram Alge-

bras

We now apply the general argument in the previous section 6.1 to prove theorems

1.0.3, 1.0.4 on page 9, which give bases for the arrow diagram algebras Avfb and Avf.

First, we summarize the definition of Avfb and Avf from section 5.6: Avfb := Dvf/Ivfb

where Dvf are descending arrow diagrams and Ivfb is the ideal generated by the descend-

ing 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 generating relations:

+

-

6T:

XII :

:=

:=

+

For convenience later on, we also add the following third order XII3-generating rela-

tion, which is a consequence of the XII-generating relation, to the set of defining gener-

ating relations of Ivf:

- := XII3 : - =

All diagrams in this section, we may omit the arrows for simplicity but use the con-

vention left skeleton segments are always understood to precede right skeleton segments.

We will prove the following lemma which immediately implies theorems 1.0.3, 1.0.4.

Lemma 6.3.1 (Basis for Afbn , Af

n). A basis of Avfb is the subset Dvf−Lvfb of descending

arrow diagrams where Lvfb is the set of all diagrams generated by illegal general chord

diagram algebra L6T:


Figure 6.2. The illegal general chord diagram algebra L6T

A basis of Avf is the subset Dvf − Lvf of arrow diagrams where Lvfb is the set of

diagrams generated by L6T above and the following general chord diagram algebras LXII ,

and LXII3.

Figure 6.3. The illegal diagrams LXII (L) and LXII3 (R)

Proof. For each of Afb and Af, we use the general argument in lemma 6.1.4 with the

partial ordering on the general chord diagram algebras constructed from an ordered set

of functions N := (NG1 , NG2 . . . NGk) as in section 6.2. For Afb, N = (NHT ), and for Af,

N = (NHT , NX , NX3), where the general chord diagram algebras HT , X, X3 are defined

below in definition 6.4.1.

Then for each case, it suffices to show the following. First, the ordered set of function

N satisfies condition 1 in lemma 6.2.3 with g1, g2, . . . gn substituted in by the generating

relations that generate the ideal, i.e. 6T for Afb and 6T,XII,XII3 for Af.Secondly,

for each overlap type between each pair of generating relations in R, there exists at

least one syzygy as in equation 6.1 such that (NG1 , NG2 . . . NGk) satisfies condition 2 in

lemma 6.2.3.The rest of the thesis is devoted to show these two statements for both Avfb

and Avfb.


6.4 Some Lemmas on Counting Embeddings of HT ,

X, X3

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 6.4.1. Define HT, X, X3, and R1 to be the following chord diagrams:

Figure 6.4. HT : the ”Head preceding tail” general chord diagram algebra. Notice

the head and the tail do not need to be immediately adjacent to one another.

Figure 6.5. X : an ”X” arrow pair general chord diagram algebra

Figure 6.6. X3 : an ”arrow crossing a pair of arrows” general chord diagram algebra.

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 6.2.1. Following section 6.2, define the

partial ordering Ovfb on Gvf to be the one induced by the function (NHT ), and the partial


III III IV

or

I II III IV

Gap: V

V

c

a b

c

a b

Figure 6.7. Some terminology: all chords a, b, c are inside the pair of gaps I − V ;

a is inside while c is outside II − IV ; c is a “right-most” chord since its ends are

the right-most on its skeleton strands.

ordering Ovf on Gvf to be the further refined partial ordering induced by the ordered set

of functions (NHT , NX , NX3).

The following counting lemmas ( 6.4.3, 6.4.3) gives sufficient conditions for most con-

ditions in lemma 6.2.5 in the specials case where the partial orderings are induced by the

functions NHT , NX , and NX3, which count the number of embeddings of the subcripted

diagram in any given diagram. It counts the number of different factorized overlapping

MCM between HT , X, or X3 and D with the same the embedding θa of D.

Here is some terminology also illustrated in figure 6.7. Ordering words such as be-

fore,after, first, last, adjacent, left, right are all w.r.t. to the orientation and ordering

of the skeleton strands; a gap in a skeleton is where an extra disjoint strand can be added

by an ordered disjoint union i.e. either before the first strand, in between two adjacent

strands, or after the last strand; a chord is short for a single-chord diagram, including

its two skeleton strands (see figure 2.2); a chord starting or ending somewhere refers

to where respectively the first or second strand of the single-chord diagram is embedded;

and a chord outside or inside a pair of gaps in a skeleton means a chord which starts

on a skeleton strand after the first gap and ends on a skeleton strand before the second

gap. See figure 6.7.


Lemma 6.4.2. let L and D be diagrams on the same skeleton S, and let G one of HT ,

X or X3. To check that NG(θa(L)) ≥ NHT (θa(D)) for all gluing operators θa = Id for

which there is an embedding of G such that θa(D) is a factorized overlapping MCM of

G and D containing D by θa, it suffices to check that the following numbers for L are

greater than or equal to that for D:

Case G= HT For each choice of a pair of gaps in the skeleton S of L and D:

1. the number of chords ending before the first gap the number of chords starting

after the second gap;

2. the number of chords starting after the second gap;

Case G= X For each choice of a pair of gaps in the skeleton S of L and D:

1. the number of chords outside the pair of gaps;

2. the number of chords inside the pair of gaps;

Case G= X3 For each choice of a pair of gaps in the skeleton S of L and D:

1. the number of chords outside the pair of gaps;

2. the number of consecutive pairs of chords inside the pair of gaps (see fig-

ure 6.6);

and for each choice of a pair of strands in S,

4. the number 1 or 0 of left-most chord on it.

Proof. First the cases G = HT and G = X. Let J be diagram any on the skeleton S such

as L or D. Any non-trivial (θa = Id) multiple θa(J) which is an overlapping MCM of J

and G must be an ordered disjoint union of J with a single-chord diagram, the single-

chord subdiagram of G not contained in the overlap. Each such ordered disjoint union

operator θa is determined simply by the choice of a pair of gaps in the skeleton S, where


the two skeleton strands of the single-chord diagram will be. Now in the case G = HT ,

for each choice of θa, i.e. each choice of a pair of gaps, the number of embeddings of HT

into θa(J) such that the diagram θa(J) with J embedded by θa is a factorized overlapping

MCM of J and HT , is the sum of the number of chords in J ending before the first gap

and the number of chords in G starting after the second gap, which equal respectively the

number of embeddings of HT where the overlap is the right and left chord of HT . And

in the case G = X, for each choice of θa, i.e. each choice of a pair of gaps, the number

of embeddings of X into θa(J) such that θa(J) with J embedded by θa is a factorized

overlapping MCM of J and HT is the sum of the number of chords in J outside the pair

of gaps and the number of chords in J inside the pair of gaps, which equal respectively

the number of embeddings of X where the overlap is the “inner” and “outer” chord of

X (see figure 6.5).

For the case G = X3, there are four different multiples of D which any overlapping

MCM of X3 and J in which X3 is not just embedded into J has a one-chord or two-

chord overlap, and MCMs with different number of chords in their overlaps have to be

different multiples θa(J) of J . For the one-chord-overlapping MCMs, there are two cases,

separated by whether the overlap is one of the chords in the consecutive pair in X3. For

the case not, MCM has to be θa(J) in which θa disjoint union a consecutive pair to J ,

determined by the choice of a pair of gaps. and this is the only factorized overlapping

MCM possibly for such θa, and the number of different embeddings of X3 for each choice

of such θa is the number of chord outside the gaps. For the case yes, θa has to be gluing

a chord to a left-most chord in J , determined by a choice of gaps to the right of which is

a left-most chord and disjoint union a single chord outside the consecutive pairs, but for

each such θa there is possibly only one embedding of X3. For the two-chord-overlapping

MCMs, again there are the cases in which the overlap includes one of the chords in the

consecutive pair in X3, or not. for the case not, the MCM θa(J) has to be a disjoint


union of a single chord to J , determined by the choice of a pair of gaps in S, with the

overlap being the consecutive pair of X3, so for each such θa the number of overlapping

MCM which contains J by θa is the number of different embedded consecutive pairs in

J inside the chosen pair of gaps. for the case yes, θa(J) has to be gluing a chord to a

left-most chord in J , the overlap is the left-most chord and the outer most chord, so

for each such θa or choice of left most chord, the number of overlapping MCM which

contains J by θb is the number of different embedded single chord pairs in J outside gaps

to immediately left of a chosen left-most chord. See figure 6.4.

Here is a further simplification:

Lemma 6.4.3. If two chord diagrams L and D on the same skeleton are the same up

to reordering of chord ends on each skeleton strand, then all the numbers lemma to be

compared between L and D except the very last one on the list are equal for L and D.

Proof. The definition of all the numbers except the last one which concerns left most

chords in J do not depend on the ordering of the chord ends on each skeleton strand.

Remark 6.4.4. It is important to select a diagram to count so the induced partial ordering

defines strict ordering relations needed for the unique leading diagrams is respected by

the−→CD operations. For example, see figure 6.9 in which L > D but θa(L) < θa(D) for

some gluing operator θa.

6.5 Grobner Argument applied to Afb

In this section, we find a basis for Afb by applying the Grobner argument in lemma 6.1.4.

Lemma 6.5.1. Let L6T be the illegal diagram in figure 6.2, and D6T be any of the

other chord diagrams in the 6T -generating relation, then Ofb defines L6T to be the unique

leading diagram of 6T-generating relations, and the definition of leading is respected by

the−→CD structure, i.e. NHT (θa(L6T )) > NHT (θa(D6T )) for all

−→CD gluing operator θa.


D:=

θ (D):=

θ (D)

a

θ (D) := b

θ (D) := c

θ (D) := b

# of MCM(XII3, D) with the

fixed embedding of D

= # of in D inside the

pair of gaps where is

added

= 2

= # of in D outside the


added next to a left-most chord

= 2

= # of in D outside the


added

= 4

= 1

Embedding of D,

Figure 6.8. Given any diagram D, e.g. the top-left diagram, there are only four

types of factorized multiples of D which can be overlapping MCMs with X3. An

example of each type is given on the left column where the dotted chords are from

D and the solid chords are added by the unary gluing operator θ. The first type

of multiplication operators ordered-disjoint unions D with a single chord diagram;

the second type glues a single-chord diagram to a left-most or right-most chord in

D; the third ordered-disjoint unions D with a pair of consecutive chords, and the

last glues a single-chord daigram to a left-most or right-most chord and also disjoint

unions another single chord to D. For each type of multiple of D, the number of

embeddings of X3 into it which give different overlapping MCM’s with D has a

simple formula and is used on the right.


L:= D:=

θ := a

θ (D) = aθ (L) = a

H<->T:=

Figure 6.9. The number NH↔T of embeddings of the diagram H ↔ T on the

top left is used for a partial ordering on diagrams. This partial ordering however is

not respected by the gluing structure since NH↔T (L) = 1 > 0 = NH↔T (D) but

NH↔T (θa(L)) = 1 < 2 = NH↔T (θa(D)) with the gluing operator θa on the top

right.

Proof. Applying lemmas 6.2.5 6.4.3, it suffices that for each pair of gaps in the three

strand skeleton of L6T or D6T , the number of chords ending before the first gap and

the number of chords starting after the second gap is greater or equal for L6T than

for D6T , and the number of embeddings of HT into L6T is strictly greater than D6T .

We enumerated these numbers in figure 6.10 below, and circled the leading diagrams

according to these number, which indeed is L6T .

Now, there is only one overlap type between two 6T generating relations, and the

overlapping factorized MCM of their leading diagrams is L6T−6T and the associated δ6T−6T

are:

Figure 6.11. 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:


6T Diagrams

III

I (in / L ; out / R )

II (in / L ; out / R )

/ ; 1 / ; 0 / ; 1

0; /1 ; /1 ; /

1 0 0 0 0 0# HT Embeddings

Figure 6.10. Number of chords ending on the left of (“in/L”) and starting on the

right of (“out/R ”) a gap (labeled “I” or “II”) and number of different embeddings

of the HT diagram in the diagrams of the 6T generating relation. E.g. the pair

“/;1” in the row “I (in/L; out/R)” under the first two diagrams (top left) says

that the number of chords ending on the left of gap I is the same for all diagrams in

the table and thus omitted (/), and the number of chords starting on the right of

gap I in the first two diagrams are the same and is 1. The diagram which has the

maxima in all rows are circled, and by lemma 6.4.3 is the well-defined unique leading

diagram of the 6T relations.


= 06T-6T :

where the sum is over the 4 ways of placing the ends of the 6T-generating relator

and for each way the 3 ways of placing the extra chord with an end not on any segment

on which the 6T-generating relator has ended, and the brackets are the “commutator”

brackets as usual; thus, there are 24 terms in total.

Finally, we need to show that the factorized overlapping MCM L6T−6T of two 6T

leading diagrams is a unique maximum among all leading diagrams appearing in the

syzygy.

Lemma 6.5.2. The diagram L6T−6T in figures 6.11 is the unique maximum chord di-

agram among the leading diagrams of the relations in the 6T − 6T syzygy w.r.t to the

partial ordering induced by NHT , and this definition of maximum is unchanged under

application of any−→CD gluing operations, i.e. NHT (θa(L6T−6T )) > NHT (θa(D6T−6T )) for

all−→CD gluing operator θa, where D6T−6T is any of the 6T relations leading diagrams other

than L6T−6T in the 6T − 6T syzygy.

Proof. The argument is the same as in lemma 6.5.1. The table of relevant numbers

to be compared is in figure 6.12 with the circled diagrams to be the unique maximum

well-defined under−→CD gluing operations.

We have now completed the proof of theorem 1.0.3.

Remark 6.5.3. Afb restricted to diagrams on only a one-strand skeleton and to gluing

operation restricted to binary concatenation is a graded associative algebra graded by

the number of chords, of dimension n! at degree n. Thus, Afb is isomorphic to ⊔∞n=1KSn

as a vector space but not as an algebra, because the multiplication using composition in

⊕∞n=1KSn is closed within each n.


I


/ ; 2 / ; 2 / ; 2

1; /2 ; /2 ; /

6T-6T Leading Diagrams III III

III

1 ; 1 0 ; 1 0 ; 1

3 1 2 1 1 1

I

II

/ ; 1 / ; 2 / ; 2

1; /2 ; /2 ; /III

1 ; 1 0 ; 1 0 ; 1

2 1 2 1 1 1

I

II

/ ; 1 / ; 2 / ; 2

1; /2 ; /2 ; /III

1 ; 0 1 ; 0 1 ; 1

1 1 1 2 1 2

(in / L ; out / R )

(in / L ; out / R )

(in / L ; out / R )(in / L ; out / R )

(in / L ; out / R )

(in / L ; out / R )(in / L ; out / R )

(in / L ; out / R )

I

II

/ ; 1 / ; 2 / ; 2

2; /2 ; /2 ; /III

1 ; 0 1 ; 0 1 ; 1

1 1 1 2 1 3

(in / L ; out / R )(in / L ; out / R )

(in / L ; out / R )

leading diagrams from

# HT Embeddings

# HT Embeddings

# HT Embeddings

# HT Embeddings



of the HT diagram in the leading diagrams of relations in the 6T-6T syzygy. E.g.

the pair “/;2” in the row “I (in/L; out/R)” under the first two diagrams on the

top right says that the number of chords ending on the left of gap I is the same for

all diagrams in the table and thus omitted (/), and the number of chords starting

on the right of gap I in the first two diagrams are the same and is 2. The diagram

which has the maxima in all rows are circled, and by lemma 6.4.3 is a well-defined

unique maximum leading diagram among all others in the 6T-6T syzygy.


6.6 Grobner Argument applied to Af

In this section, we find a basis for Af by applying the Grobner argument in lemma6.1.4.

6.6.1 Well-definedness of Leading Terms of Relations

Lemma 6.6.1. Let L6T , LXII , LXII3 be as in figures 6.2, 6.3, and D6T , DXII , DXII3

be any of the other chord diagrams in the 6T -, XII-, and XII3-generating relations

respectively, the the partial ordering induced by the ordered set of counting functions

(NHT , NX , NX3) gives L6T , LXII and LXII3 as the unique leading diagrams for the 6T -,

XII- and XII3- generating relations, and these definitions of leading diagrams respect

the−→CD structure, i.e.

For 6T NHT (θa(L6T )) > NHT (θa(D6T )) ∀ θa;

For XII NHT (θa(LXII)) = NHT (θa(DXII)) ∀ θa, NX(θa(LXII)) > NX(θa(DXII)) ∀ θa;

For XII3 NHT (θa(LXII3)) = NHT (θa(DXII3)) ∀ θa, NX(θa(LXII3)) = NX(θa(DXII3)) ∀ θa,

NX3(θa(LXII3)) = NX3(θa(DXII3)) ∀ θa.

Proof. It was already shown above in lemma 6.5.1 that L6T is a well-defined leading

diagram of the 6T -generation relation. Now, since both diagrams in the XII-relation

are the same up to reordering of chord ends on the same skeleton strands, by lemmas 6.2.5,

6.4.3, and 6.4.3, it suffices to that the number of embeddings of X into LXII is strictly

bigger than that into DXII . Again, both diagrams in the XII3-relation are the same up

to reordering of chord ends on the same skeleton strands, and both contain no right-most

or left-most chords, then by the same lemmas ( 6.2.5, 6.4.3, and 6.4.3), it suffices that

the number of embeddings of X3 into LXII3 is strictly bigger than that into DXII3.


6.6.2 Enumeration of Overlap Types and Syzygies

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. We first list the ones among the XII- and XII3-generating relations,

and then the ones which involves also 6T .

XII-XII: There is only one overlap type and the overlap diagram LXII−XII and the

associated δXII−XII are as follow:

Figure 6.13. δXII−XII (L) ; LXII−XII (R)

-

The associated syzygy XII-XII is by construction as shown in figure 6.3.

XII3-XII3: There is no MCM of XII3-XII3 which overlaps

XII-XII3: There is only one overlap type and its overlap diagram LXII−XII3 and

associated δXII−XII3 are as follow:

Figure 6.14. δXII−XII3 (L) ; LXII−XII3 (R)

-

the associated syzygy XII-XII3 is:

- - - = 0XII-XII3 :

Here are the syzygies between 6T and one of XII and XII3:


6T-XII: There are a left and a right overlap MCM types with respective overlap

diagrams L6T−XII(L) and L6T−XII(R) and associated δ6T−XII(L) and δ6T−XII(R) as follow:

Figure 6.15. δ6T−XII(L) (L) ; L6T−XII(L) (R)

-

Figure 6.16. δ6T−XII(R) (L) ; L6T−XII(R) (R)

-

The syzygies 6T-XII(L) and 6T-XII(R) associated to the respective overlap types are

respectively the equalities on the left and the right below:

+ = _ _

= +

L L.H.S. - MID. R R.H.S. - MID.= 0 = 06T-XII (L) 6T-XII (R): ::= :=

where the short-hand notations are as follow:

: +:= _ _ := +

6T-XII3: Again, there are a left and a right overlap types with respective overlap

diagrams LXII3−6T and L6T−XII3 and associated δ6T−XII3(L) and δ6T−XII3(R) as follow:


Figure 6.17. δ6T−XII3(L) (L) ; L6T−XII3(L) (R)

-

Figure 6.18. 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 + = 0

L 6T-XII3 (L) :

where the trivial 6T-XII syzygy is

:= -

+ - + + - -

0L

and the following generating relation appears twice and has been canceled:

and the following difference of generating relations has been rewritten:

- =

The right syzygy is exactly analogous and these syzygies are of the overlap types in

figure 6.16.


6.6.3 Well-Definedness of Maximum Leading Diagrams of Syzy-

gies

Lemma 6.6.2. Each of the overlap diagrams L6T−6T , LXII−XII , L6T−XII(L), L6T−XII(R),

L6T−XII3(L), L6T−XII3(R), LXIIXII3, in figures 6.11, 6.13, 6.15, 6.16, 6.17, 6.18, and 6.14

respectively, is the unique maximum chord diagram among the leading diagrams of all the

relations in the respective syzygy w.r.t to the partial ordering induced by (NHT , NX , NX3).

The details are as follow.

First, the partial ordering induced by NHT does not separate any of the leading di-

agrams in the syzygies not involving 6T , but gives the following non-unique maximum

leading diagrams for the syzygies involving 6T :

6T-XII(L/R) L6T−XII(L) ( resp. L6T−XII(R)) in figure 6.15 ( resp. 6.16) and LHT6T−XII(L)

( resp. LHT6T−XII(R)) below:

L6T-XII (L)

HT

L =

L6T-XII (R)

HT

L =

6T-XII3(L/R) L6T−XII3(L) ( resp. L6T−XII3(R)) in figure 6.17 ( resp. 6.18) and LHT6T−XII3(L)

( resp. LHT6T−XII3(R)) below:

L6T-XII3 (L)

HT

L =

L6T-XII3 (R)

HT

L =

Secondly, the more refined partial ordering induced by (NHT , NX) does not separate

yet the leading diagrams of the relations in XII − XII3 nor between the two pairs of

maximum (w.r.t to NHT ) leading diagrams L6T−XII3(L), LHT6T−XII3(L) and


L6T−XII3(R), LHT6T−XII3(R) for the 6T−XII3(L) and 6T−XII3(R) respectively, but gives

the overlap diagrams L6T−XII(L), L6T−XII(R), and LXII−XII in figures 6.15, 6.16 and

6.13 as the unique maximum leading diagrams for the syzygies 6T −XII(L), 6TXII(R),

and XII −XII respectively.

Finally, the most refined partial ordering induced by (NHT , NX , NX3) separates the

pairs L6T−XII3(L), LHT6T−XII3(L) and L6T−XII3(R), L

HT6T−XII3(R), and gives the overlap

diagrams L6T−XII3(L), L6T−XII3(R), as well as LXII−XII3, in figures 6.15, 6.16, and

6.14 respectively, as the unique maximum leading diagrams for syzygies 6T −XII3(L),

6TXII3(R), and XII −XII3 respectively.

Proof. For 6T −6T already done in lemma and does not change by any refinement of the

ordering. t as in lemma 6.6.2, Using lemmas, it suffices to compare the numbers and also

the number of embeddings of HT into the different leading diagrams. These numbers for

the syzygies which involve 6T are listed in figure ??, and the maximum leading diagrams

we circled according to these numbers are exactly the overlap diagrams L6T−XII3(L) (resp.

L6T−XII3(R)) as between for the syzygies among XII XIIs, all leading diagrams for each

syzygy is the same up to reordering of chord-ends on the same skeleton strands, and since

no HT can be embedded in any of the leading diagrams in these syzygies, the partial

ordering induced by NHT do not separate them.

We apply the more refined partial ordering NX to syzygies with no unique maximum

yet. the diagrams in allXII−XII3 and L6T−XII3(L), LHT6T−XII3(L) and L6T−XII3(R), L

HT6T−XII3(R)

are all the same up to reordering of chord ends, and have the same number of embeddings

of X into them, so they are still not separated. But for the pairs L6T−XII(L), LHT6T−XII(L)

and L6T−XII(R), LHT6T−XII(R), even though all the same up to reordering of chord ends,

the number of embeddings of X into the overlap diagrams L6T−XII(L), and L6T−XII(R)

are strictly bigger, making them the unique maximum leading terms in 6T − XII(L),

6TXII(R). For exactly the same reasons, the overlap diagram LXII−XII in figure 6.13 is


I


/ ; 1

6T-XII(L) Leading Diagrams III

2 ; /2

I

II

(in / L ; out / R )

(in / L ; out / R )

(in / L ; out / R )

Diagrams

from 6T's

Diagrams

from XII's

/ ; 1

1 ; /1 2 1 1 1 1 1

/ ; 1

2 ; /

2

/ ; 0

1 ; /

0 0 0 0 0 0 0

/ ;0

2 ; /

/ ; 1

1 ; /# HT Embeddings

# HT Embeddings



of the HT diagram in the leading diagrams of relations in the 6T-XII trivial syzygy.

E.g. the pair “/;1” in the row “I (in/L; out/R)” under the first four diagram says

that the number of chords ending on the left of gap I is the same for all diagrams

in the table and thus omitted (/), and the number of chords starting on the right

of gap I in the first four diagram is 1. The three diagrams (two of which are the

diagram) with the maximum numbers in all its rows are circled, and by lemma 6.4.3

are the well-defined maximum leading diagrams among all others in the 6T-XII(L)

syzygy.

the unique maximum leading diagrams for the syzygies XII −XII respectively.

Finally, again for exactly the same reasons, (that the leading diagrams to be com-

pared are the same up to reordering of chord-ends on the same strands but the embedding

of the diagram defining the ordering is strictly bigger for one diagram), NX3 separates

the pairs L6T−XII3(L), LHT6T−XII3(L) and L6T−XII3(R), L

HT6T−XII3(R), and gives the over-

lap diagrams L6T−XII(L), and L6T−XII(R), as well as LXII−XII3 as the unique maximum

leading terms in 6T −XII(L), 6TXII(R), and XII −XII3 respectively.


I


/ ; 1

6T-XII Trivial Leading Diagrams

3 ; /

I

II

/ ; 1 / ; 0 / ; 1

3 ; / 3 ; / 2 ; /2 0 0 2 2

(in / L ; out / R )

(in / L ; out / R )

(in / L ; out / R )

III

Diagrams

from 6T's

Diagrams

from XII's

3

/ ; 1

3 ; /

3

Diagrams

fromXII3

# HT Embeddings

# HT Embeddings



of the HT diagram in the leading diagrams of relations in the 6T-XIIL trivial syzygy,

and in the leading diagram (top right)of a XII3 relation which is the sum of of two

XII relations, one from the 6T-XIIL syzygy and one from the 6T-XIIL trivial syzygy.

E.g., the pair “/;1” in the row “I (in/L; out/R)” under the second diagram (top

middle) says that the number of chords ending on the left of gap I is the same for all

diagrams in the table and thus omitted (/), and the number of chords starting on

the right of gap I in that diagram is 1. The first diagrams in both rows are crossed

out because the relations of which they are leading diagrams do not appear in the

6T-XII3(L) syzygy. These relations of are either cancelled or combined with another

relation. The diagrams with the maximum number in all its rows are circled, and by

lemma 6.4.3 is the well-defined maximum leading diagrams among all others in the

6T-XII3(L) syzygy.

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