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Flat Virtual Pure Tangles
by
Karene Chu
A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics
University of Toronto
Copyright c⃝ 2012 by Karene Chu
Abstract
Flat Virtual Pure Tangles
Karene Chu
Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
2012
Virtual knot theory, introduced by Kauffman [Kau], is a generalization of classical
knot theory 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
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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.
Chapter 1. Introduction 5
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.
Chapter 1. Introduction 6
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
Chapter 1. Introduction 7
:= :=_ _
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,
Chapter 1. Introduction 8
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
Chapter 1. Introduction 9
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.
Chapter 1. Introduction 10
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.
Chapter 1. Introduction 11
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.
Chapter 1. Introduction 12
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
Chapter 2. Preliminaries 15
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.
Chapter 2. Preliminaries 16
-
++
+
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.
Chapter 2. Preliminaries 17
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.
Chapter 2. Preliminaries 18
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.
Chapter 2. Preliminaries 19
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
Chapter 2. Preliminaries 20
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.
Chapter 2. Preliminaries 21
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).
Chapter 2. Preliminaries 22
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.
Chapter 2. Preliminaries 23
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
Chapter 2. Preliminaries 24
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.
Chapter 2. Preliminaries 25
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:
Chapter 2. Preliminaries 26
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.
Chapter 2. Preliminaries 27
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.
Chapter 2. Preliminaries 28
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
Chapter 2. Preliminaries 29
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.
Chapter 2. Preliminaries 30
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.
Chapter 2. Preliminaries 31
+
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
Chapter 2. Preliminaries 32
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.
Chapter 2. Preliminaries 33
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 .
Chapter 2. Preliminaries 34
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.
Chapter 2. Preliminaries 35
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 38
+ + +
- -
-
+ +
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 39
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 40
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
Chapter 3. Classification of Pure Descending Virtual Tangles 41
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 42
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 43
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
Chapter 3. Classification of Pure Descending Virtual Tangles 44
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,
Chapter 3. Classification of Pure Descending Virtual Tangles 45
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:
Chapter 3. Classification of Pure Descending Virtual Tangles 46
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 47
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 48
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 49
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.
Chapter 3. Classification of Pure Descending Virtual Tangles 50
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,
Chapter 3. Classification of Pure Descending Virtual Tangles 51
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
Chapter 3. Classification of Pure Descending Virtual Tangles 52
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 ⟩
Chapter 3. Classification of Pure Descending Virtual Tangles 53
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:”
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams56
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.
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams57
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.
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams58
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.
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams59
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
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams60
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.
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams61
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.
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams62
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
Chapter 4. Preliminaries: Minimal Common Multiples of Chord Diagrams63
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
.
Chapter 5. Preliminaries: the Associated Graded Spaces 66
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 .
Chapter 5. Preliminaries: the Associated Graded Spaces 67
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.
Chapter 5. Preliminaries: the Associated Graded Spaces 68
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.
Chapter 5. Preliminaries: the Associated Graded Spaces 69
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.
Chapter 5. Preliminaries: the Associated Graded Spaces 70
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
Chapter 5. Preliminaries: the Associated Graded Spaces 71
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.
Chapter 5. Preliminaries: the Associated Graded Spaces 72
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:
Chapter 5. Preliminaries: the Associated Graded Spaces 73
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.
Chapter 5. Preliminaries: the Associated Graded Spaces 74
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.
Chapter 6. Proof of Bases of Afb and Af 77
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
Chapter 6. Proof of Bases of Afb and Af 78
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.
Chapter 6. Proof of Bases of Afb and Af 79
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) | .
Chapter 6. Proof of Bases of Afb and Af 80
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,
Chapter 6. Proof of Bases of Afb and Af 81
∀ 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.
Chapter 6. Proof of Bases of Afb and Af 82
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.
Chapter 6. Proof of Bases of Afb and Af 83
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
Chapter 6. Proof of Bases of Afb and Af 84
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:
Chapter 6. Proof of Bases of Afb and Af 85
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.
Chapter 6. Proof of Bases of Afb and Af 86
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
Chapter 6. Proof of Bases of Afb and Af 87
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.
Chapter 6. Proof of Bases of Afb and Af 88
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
Chapter 6. Proof of Bases of Afb and Af 89
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
Chapter 6. Proof of Bases of Afb and Af 90
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.
Chapter 6. Proof of Bases of Afb and Af 91
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
pair of gaps where is
added next to a left-most chord
= 2
= # of in D outside the
pair of gaps where is
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.
Chapter 6. Proof of Bases of Afb and Af 92
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:
Chapter 6. Proof of Bases of Afb and Af 93
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.
Chapter 6. Proof of Bases of Afb and Af 94
= 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.
Chapter 6. Proof of Bases of Afb and Af 95
I
II (in / L ; out / R )
/ ; 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
Figure 6.12. 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 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.
Chapter 6. Proof of Bases of Afb and Af 96
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.
Chapter 6. Proof of Bases of Afb and Af 97
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:
Chapter 6. Proof of Bases of Afb and Af 98
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:
Chapter 6. Proof of Bases of Afb and Af 99
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.
Chapter 6. Proof of Bases of Afb and Af 100
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
Chapter 6. Proof of Bases of Afb and Af 101
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
Chapter 6. Proof of Bases of Afb and Af 102
I
II (in / L ; out / R )
/ ; 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
Figure 6.19. 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 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.
Chapter 6. Proof of Bases of Afb and Af 103
I
II (in / L ; out / R )
/ ; 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
Figure 6.20. 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 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.
Bibliography
[BEER] L. Bartholdi, B. Enriquez, P. Etingof, E. Rains Groups and Lie algebras
corresponding to the Yang-Baxter equations, J. Algebra 305 (2006), no. 2, 742–
764,arXiv:math/0509661.
[BN1] D. Bar-Natan, u, v, and w-Knots: Topology, Combinatorics and Low and High
Algebra, .
[BN2] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472.
[BN3] D. Bar-Natan, Finite Type Invariants, in Encyclopedia of Mathematical Physics,
(J.-P. Francoise, G. L. Naber and Tsou S. T., eds.) Elsevier, Oxford, 2006 (vol. 2 p.
340).
[BN4] D. Bar-Natan, Finite Type Invariants of W-Knotted Ob-
jects: From Alexander to Kashiwara and Vergne, in preparation,
http://www.math.toronto.edu/~drorbn/papers/WKO/.
[BHLR] D. Bar-Natan, I. Halacheva, L. Leung, and F. Roukema, Some
Dimensions of Spaces of Finite Type Invariants of Virtual Knots,
http://www.math.toronto.edu/~drorbn/papers/v-Dims/, arXiv:0909.5169.
[BS] D. Bar-Natan and A. Stoimenow, The fundamental theorem of Vassiliev invariants,
in Proc. of the Arhus Conf. Geometry and physics, (J. E. Andersen, J. Dupont,
104
Bibliography 105
H. Pedersen, and A. Swann, eds.), lecture notes in pure and applied mathematics
184 (1997) 101–134, Marcel Dekker, New-York. Also arXiv:q-alg/9702009.
[EK] P. Etingof and D. Kazhdan, Quantization of Lie Bialgebras, I, Selecta Mathematica,
New Series 2 (1996) 1–41, arXiv:q-alg/9506005.
[GPV] M. Goussarov, M. Polyak and O. Viro, Finite type invariants of classical and
virtual knots, Topology 39 (2000) 1045–1068, arXiv:math.GT/9810073.
[Hav] A. Haviv, Towards a diagrammatic analogue of the Reshetikhin-Turaev link invari-
ants, Hebrew University PhD thesis, September 2002, arXiv:math.QA/0211031.
[Kau] L. H. Kauffman, Virtual Knot Theory, European Journal of Combinatorics 20
(1999) 663–690, arXiv:math.GT/9811028.
[Ku] G. Kuperberg, What is a Virtual Link, Algebraic and Geometric Topology 3 (2003)
587591, arXiv:math.GT/0208039.
[Ma1] V. O. Manturov, On Free Knots, arXiv:0901.2214.
[Ma2] V. O. Manturov, On Free Knots and Links, arXiv:0902.0127.
[NS1] S. Naik and T. Stanford, A Move on Diagrams that Generates S-Equivalence of
Knots, Journal of Knot Theory and its Ramifications 12 (2003), no. 5, 717-724.
[Pol] M. Polyak, On the Algebra of Arrow Diagrams, Letters in Mathematical Physics
51 (2000) 275–291.