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CMS Books in Mathematics
Canadian
Mathematical Society
David M. Jackson Société mathématique
du Canada
Iain Moffatt
An Introduction
to Quantum
and Vassiliev
Knot Invariants
CMS Books in Mathematics
Editor-in-Chiefs
Karl Dilcher, Halifax, Canada
Keith Taylor, Halifax, Canada
Series Editors
Leah Edelstein-Keshet, Vancouver, Canada
Niky Kamran, Montreal, Canada
Mikhail Kotchetov, St. John’s, Canada
Martin Barlow, Vancouver, Canada
Heinz Bauschke, Kelowna, Canada
CMS Books in Mathematics is a collection of graduate level textbooks and
monographs published in cooperation with the Canadian Mathematical Society
since 1999. This series offers authors the joint advantage of publishing with a major
mathematical society and with a leading academic publishing company. The series
is edited by Karl Dilcher and Keith Taylor. The more than 20 titles already in print
cover such diverse areas as combinatorics, number theory, optimization, dynamical
systems and the history of mathematics, to name a few. Books in this series will
appeal to all mathematicians, whether pure or applied, whether students or
researchers.
An Introduction to Quantum
and Vassiliev Knot Invariants
123
David M. Jackson Iain Moffatt
Faculty of Mathematics Department of Mathematics
University of Waterloo Royal Holloway, University of London
Waterloo, ON, Canada Egham, UK
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
This book grew from a series of discussions between its authors on the combina-
torics of Vassiliev and quantum knot invariants. We were fascinated both by the
area itself, and by the rich interaction between the combinatorial and algebraic ideas
that supported it. Moreover, we recognised that the area perfectly exhibited many
of the important stages that are passed through in the development of a mathe-
matical theory. Such stages may so easily remain hidden when a reader is presented
with a complete and polished theory, yet experiencing the thinking behind the
transitions from one stage to another is vital for the aspiring mathematician.
Three key themes stood out in these discussions.
• The first theme is the constructive approach, which is fundamental to mathe-
matics in general, and to algebraic combinatorics in particular. It is an approach
in which the preliminary step to handling and understanding a difficult object
(for us, a knot) is to derive a description of it as a combinatorial structure. This
can open the way to decompose the structure into ‘elementary pieces’ that may
then be described in algebraic terms. We may then hope to work backwards to
associate an algebraic structure with the original object.
• The second theme is how one mathematical object may be transformed into one
more amenable to further investigation, while still ensuring that key information
(for us, the isotopy class of a knot) in the former is retained in the latter.
• The third theme is the use of mathematical techniques for working effectively
within the algebras that arise in these processes (e.g. ribbon Hopf algebras). In
doing so the intention is, of course, to elicit the key information (e.g. the isotopy
class of a knot) that has been retained in the more amenable object, once the
association with the original object (e.g. a knot) has been established.
Given that the area of knot theory perfectly illustrates the use of these mathe-
matical processes that are often hidden from students, we felt that an introduction to
this theory, and particularly to its interactions between algebra, combinatorics and
topology, could serve as an excellent step in the mathematical development of a
student. This thinking led to perhaps an unusual expository aim in writing this
book. Rather than writing a text that serves primarily as an introduction to quantum
v
vi Preface
knot theory (although we certainly hope that we have provided a worthy intro-
duction to that theory), we wanted to reveal mathematical approaches available for
engaging with a difficult mathematical object. Accordingly, we have adopted the
following expository principles:
1. to focus on the interplay between combinatorics, algebra, and topology;
2. to take a workmanlike approach to the abstract material, and to the acquiring of
a facility with, for example, calculations within the algebras that arise;
3. to write a book to be read, rather than to serve primarily as a reference volume;
4. to use exercises to aid learning rather than to test;
5. to write a book that will serve a broad spectrum of aspiring mathematicians, and
not solely those who wish to pursue knot theory;
6. to offer an aspiring knot theorist a grounding in the fundamentals of Vassiliev
and quantum knot invariants, thereby enabling a reader to successfully tackle the
more specialised literature;
7. and, finally, to keep the volume to a manageable length.
Writing to these principles has, of course, affected what content and which topics
we have included or omitted. For example, an expert reader will undoubtedly notice
that we have chosen not to give a full development of the theory of quantum groups
or monodromy solutions of the Knizhnik–Zamolodchikov equations, nor do we
discuss Chern–Simons theory. While these topics are beautiful and central to the
area, a proper exposition of them takes us too far from our guiding expository
principles. For similar reasons, in a few places we decided to quote results without
proof. We also felt we could safely omit these topics since they are served by other
excellent texts in the area, such as [37, 90, 144]. In fact, the present volume resulted
from paring down a much longer volume. This was a very painful process, par-
ticularly when it involved excising wonderful pieces of mathematics, but we believe
that the resulting text better serves the broader mathematical community.
In writing this book, our emphasis has been on the use and the flow of math-
ematical ideas. We have tailored our exposition to an advanced undergraduate or
beginning graduate student, who may intend to pursue a research area other than
topology or to a researcher from another field who wishes to have a rapid intro-
duction to the area. In this volume, the reader will see versatile general results, such
as isomorphism theorems, and how they may be used in practice; will develop an
appreciation of how various types of algebras arise, and how the introduction of
algebraic objects is driven by application; will see how a profound theory can be
built based on the tools acquired in undergraduate studies; and will develop a
proficiency in a variety of algebraic, combinatorial, and topological techniques that
can be applied in areas beyond the scope of this book.
Finally, we hope that the reader will be captivated by the beauty of the area and
fascinated by its algebraic combinatorial nature, as indeed we were.
In preparing this book, we have benefited from and enjoyed conversations with
and support from many people. It is impossible to thank each person by name,
although special thanks are due to P. Etingov, S. Garoufalidis, S. Lando, and
Preface vii
M. Perry for valuable discussions at various stages. Special thanks are due to our
spouses Jennifer and Vikki for their continuing support and patience through this
project.
ix
x Contents
xiii
xiv Frequently Used Notation
The problem we are interested in is the classical problem in knot theory of deciding
whether two knots are equivalent. We start intuitively by thinking of a knot as the
familiar configuration that results from looping a length of string around itself and
threading an end through the loops so formed, as shown below:
xv
xvi Introduction
For two such knots K and K 0 ; there is no longer the possibility of unknotting K and
then retying it as K 0 since there are no loose ends, as there were for the case of a
knotted line.
Intuitively, we regard two knots K and K 0 as being equivalent if we can move the
circles of string around in space until the knots look the same. (As we shall see, this
idea is formalised by requiring an orientation preserving homeomorphism of R3
that takes one knot to the other.) We are interested in the question of whether two
knots are equivalent or whether they are not.
By forming them out of a piece of string, for example, the reader will quickly
convince himself or herself that the following two drawings represent the same knot
(i.e. we can deform one so it looks like the other).
Also by forming them out of a piece of string, say, the reader should also be able
to convince himself or herself that the following two drawings represent different
knots.
Although the reader will be convinced that these two knots cannot be changed
into one another, it is at least theoretically possible that there is some way to do it
using a trick that he or she did not spot. Thus, we need a way to prove that no
such trick exists and that the knots can never be changed into one another.
(In Exercise 2.26, we shall see that these two knots are indeed different.)
Deciding whether two drawings represent different knots is a hard problem. As
an illustration of the difficulty, the following two knots, called the Perko Pair, were
thought to be distinct until the 1970s when Perko [148] showed that they are not.
Introduction xvii
Exercise 1 Verify that the two drawings above do indeed represent the same
knot.
Deciding whether knots are inequivalent requires the use of a knot invariant.
A knot invariant is a function h from knots to some set (typically a set of numbers,
polynomials, or groups) with the property that if K and K 0 are knots, then
hðKÞ 6¼ hðK 0 Þ implies that K and K 0 are different knots.
The intent, of course, is to find a set S (e.g., the set of complex numbers, or a set of
polynomials) such that testing for hðKÞ 6¼ hðK 0 Þ in S is easier than testing that
K 6¼ K 0 in the set of all knots. Constructing computable and effective knot
invariants is difficult. In this book, we describe the construction of two related
families of knot invariants called Quantum Invariants and Vassiliev invariants.
It is instructive to reflect upon progress in knot theory during the last two hundred
and fifty years. The first appearance of knots in a mathematical context was in a
paper by Vandermonde in 1771, followed by some notes on knots by Gauss in 1794
that were little more than collections of drawings. Fifty years later the first systematic
attempts to study and classify knots, due in the most part to Tait, Kirkman, and Little,
although substantial, were thwarted by the absence of a suitable mathematical
framework to support them. Quite simply, although these early proponents of knot
theory were often able to show that two knots were the same, they were often unable
to show that two knots were different. This changed with the introduction of alge-
braic topology in the early twentieth century, and mathematicians such as Dehn,
Seifert, Reidemeister, Alexander, and Wirtinger were attracted to the subject.
Connections between knots and the topology of 3-manifolds were discovered.
In the 1980s, a revolution in knot theory was sparked through the work of Jones.
This resulted in a new class of knot invariants called Quantum Invariants, and
connections were established with statistical mechanics, quantum physics, quantum
groups, and Lie algebras. Drinfel’d, Jones, and Kontsevich each received the Fields
Medal, the highest award in mathematics, for work that contributed to knot theory.
It is this family of knot invariants that we focus upon here. (For a more detailed
history of knot theory, we refer the reader to [153].)
We don’t mention Vassiliev invariants.
Before plunging into the mathematics, we briefly describe the structure of this
book.
Part I provides an overview of Basic Knot Theory. Its purpose is to familiarise the
reader with the basic concepts of knot theory that are used in the rest of the book.
xviii Introduction
Part I
Foundational
knot theory
Ch. 1 – 3
Braids
Ch. 4
Part II Part III Part IV
Part II
Ribbon Hopf
Jacobi diagrams q-tangles
algebras
Ch. 13 Ch. 15
Ch. 8
Reshetikhin-
Lie algebras Jacobi diagrams
Turaev invariants
Sec. 8.4.1 Ch. 13
Ch. 9
Kontsevich
Invariant
Ch. 17
Universality
Ch. 18
Exercises can be found throughout the book. These exercises are formative
rather than summative, written with the intention of developing the reader’s
understanding as he or she progresses through the book. As such, exercises are
positioned within the main chapter text, rather than collected at the end of chapters.
They also form part of the narrative. A collection of hints for the exercises can be
found towards the end of the book. Appendix A contains the prerequisites on linear
algebra and module theory required for the text. The remaining appendices contain
details of algebraic computations that it is important to work through, but have been
moved to the appendix so as not to distract from the development of the theory. The
bibliography includes all the mathematical sources we consulted to prepare this
book. A list of frequently used notation can be found after the table of contents, on
page xiii.
Part I
Basic Knot Theory
Chapter 1
Knots
We assume that the reader is familiar with basic topology but, for convenience, we
recall that a homeomorphism is a continuous map with a continuous inverse, and
that a function is an embedding if it is a homeomorphism onto its image. All the
topologies we consider here are those inherited from the standard topology of Rn .
Note that throughout the book, we work in the piecewise-linear setting, rather than
the smooth one. In particular all embeddings are piecewise-linear.
The physical idea of a knot, that was stated in the introduction as a “knotted circle
of string”, has a mathematical idealisation encapsulated by the following definition.
01 31 41 10157
Drawings of a few knots are given in Fig. 1.1 together with their standard symbols.
The knot 01 is called the unknot, 31 is the trefoil, and 41 is the figure-of-eight knot.
The symbols such as 31 used here refer to the names of the knots in the standard
Rolfsen tables [158].
the first link is called the Hopf link, and the second the Borromean rings.
Note that no embedded circles in a link can intersect and that a knot is precisely a
one-component link. More properly, knot theory is the study of links, and it is com-
mon practice in the area to prefix a term by “knot” when it should really be prefixed
by “link”. Usually, but not exclusively, the term “knot invariant”, for example, is used
to refer to invariants of links as well as knots. This standard abuse of terminology
within the discipline should be kept in mind when reading any material about knot
theory.
Knots and links are considered up to a relation called ambient isotopy. This relation
coincides with the intuitive notion of deforming a knotted circle in three-dimensional
space without cutting it or gluing parts of it together. The formal definition is the
following.
Definition 1.3 (Ambient isotopy of links). Two knots (or links) K and K are
said to be (ambient) isotopic if there is a family of homeomorphisms h t : R3 →
R3 , t ∈ [0, 1], such that h 1 (K ) = K , h 0 = idR3 and (x, t) → (h t (x), t) defines a
homeomorphism from R3 × [0, 1] to itself.
The idea behind this definition is that two knots are equivalent if R3 can be continu-
ously deformed in a way that takes one knot to the other.
We shall often write “isotopy” for “ambient isotopy”, but it should be understood
that all isotopies considered here are ambient isotopies.
Definition 1.4 (Equivalence of links). Two knots or links are equal or equivalent
if they are isotopic. An equivalence class of the set of knots or links modulo isotopy
is called an isotopy class.
K1 K2 K3 K4
6 1 Knots
However, the knots K 3 and K 4 are not equivalent (See Exercise 2.26).
p1 pk ··· pn
(2, 3, −5, 4) =
There is an inherent difficulty in showing that knots are not equivalent. On the
one hand, we can demonstrate that two knots or links are equivalent by describing a
sequence of deformations taking one to another, as in the first part of Example 1.5.
On the other hand, being unable to find such a sequence of deformations does not
necessarily imply that such a sequence cannot exist—perhaps we did not look hard
enough for one. Instead, we need to prove that no sequence of deformations can exist.
1.1 Knots and Equivalence 7
This requires the use of a knot invariant. That is to say, we need to find a (sufficiently
discriminating) knot invariant whose evaluations on the two knots are different. We
shall discuss this topic in Chap. 2.
With minor abuse of language, it is usual in knot theory to use the term “knot” to
mean (i) an embedding of S1 into R3 , (ii) the image of an embedding under ambient
isotopy, or (iii) an equivalence class of embeddings under ambient isotopy, and to
do similarly for the term “link”. This should not cause any difficulties since any
ambiguity is resolved by the context.
Here (and in much of knot theory literature) we shall avoid pathologies and con-
sider only knots and links that correspond closely to the intuitive notion of a knotted
string. To do this, we need the following definition.
Definition 1.7 (Tame link). A link is said to be tame if it is ambient isotopic to a set
of simple closed polygons in R3 whose sides are straight line segments; otherwise it
is said to be wild.
The tameness condition excludes infinite limiting processes and ensures that we are
studying knots and links that conform to the intuitive idea of a knot or link. Wild
knots, on the other hand, may have bizarre and counter-intuitive properties and,
although the existence of such knots has been known since the 1940s, remarkably
little is known about them. We shall be concerned exclusively with tame knots and
links and, henceforth, whenever we use the term “knot” or “link” we always mean a
tame one.
Links whose images are polygons will be referred to as polygonal links and, since
we are restricting attention to tame links and consider links up to isotopy, we can
and shall assume that links are polygonal, and therefore amenable to the techniques
of piecewise-linear topology. (For background on piecewise-linear topology, see, for
example [162].) It will be seen that this provides a combinatorial formulation of
ambient isotopy.
Our convention throughout is to draw links, link diagrams and other knotted
objects as smooth curves. Nevertheless, the reader should regard all of these smooth
curves as a polygonal curves whose vertices and edges are “so small that the curves
appear smooth” (as is the case with any pixellated image on a computer screen). This
is recorded for reference purposes by the following.
Convention 1.8. Links, link diagrams and other knotted objects are drawn here as
smooth curves but are to be regarded as polygonal curves.
The advantage of realising links as polygonal links, and so stepping into the world
of piecewise-linear topology, is that we can consider them as combinatorial objects.
In particular, we can obtain a combinatorial reformulation of ambient isotopy as
follows, where ∂(T ) denotes the boundary of the solid triangle T. Such a triangle is
represented here by shading its interior.
Definition 1.9 (Δ- and Δ−1 -moves). Let L be a polygonal link, let p be a straight
line segment of L, and let T be a solid triangle in R3 such that L ∩ T = p and
∂(T ) = p ∪ q ∪ r . Then a Δ-move is a move which replaces the edge p of L with
q ∪ r of T . The inverse of this process is called a Δ−1 -move.
8 1 Knots
q r q r
T T
p p
Δ−1
Since T is solid, no straight line segment of L may pass through it. Thus, in other
words, the Δ-move replaces the edge p by q ∪ r provided no edge of the link passes
through the solid triangle T that is used in the construction.
The following theorem gives equivalent formulations of knot equivalence. We
exclude its proof and refer the interested reader to [31] for one.
Theorem 1.10. Let L and L be (tame) links in R3 . Then the following are equiva-
lent:
1. L and L are ambient isotopic.
2. There is an orientation preserving piecewise-linear homeomorphism h : R3 →
R3 such that h(L) = L .
3. There is a finite sequence of Δ- and Δ−1 -moves taking (a polygonal representative
of) L to (a polygonal representative of) L .
Instead of working with knots and links as objects in R3 , we can work with drawings
of them on the plane R2 (in effect we have been doing so above!). We can do this
by considering the images of knots and links in R3 under sufficiently nice (they are
termed regular) projections from R3 onto R2 . It is convenient for some arguments
to fix a rectangular Cartesian coordinate system in the ambient space R3 and to
project onto a fixed plane (typically the plane z = 0 in which case the projection
maps (x, y, z) to (x, y, 0)). We adopt the following convention.
Convention 1.11. The rectangular Cartesian coordinate system, O x yz, is chosen
such that all links lie on the same side of the plane of projection.
There may be points of a knot that are mapped by projection to the same point
in the plane z = 0. Such a point is called a multiple point. To make sure that our
diagrams do not hide structure of the link, or suggest structure that is not there, we
need to insist that all multiple points are “nice” in the following sense.
1.2 Knot and Link Diagrams 9
Fig. 1.2 Permissible (a) and impermissible (b, c, d) multiple points of a regular projection
Fig. 1.3 A knot (a) and link (c) (drawn using [169]) and their corresponding diagrams (b, d)
that the crossings alternate between over- and under-crossings in a tour along the
component. Such a knot diagram is said to be an alternating diagram. Similarly, a
link diagram is alternating if in a tour along each component the crossings alternate
between over- and under-crossings. An alternating knot or link is one that admits an
alternating diagram.
Note that there is a subtlety in this definition for it states that a knot or link is alter-
nating if it has an alternating diagram. An alternating knot will also have infinitely
many diagrams that are not alternating. Thus, if we are given a non-alternating knot
diagram, it may or may not represent an alternating knot.
Exercise 1.15. Show that the following non-alternating diagram represents an alter-
nating knot.
Diagrams also enable us to find easy descriptions of some classes of knots and
links.
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