Knots, Links and Their Invariants: An Elementary Course in Contemporary Knot Theory [1 ed.] 1470471515, 9781470471514, 9781470473112

Table of contents :
Front Cover
Half Title
Title
Copyright
Contents
Foreword
Permissions & Acknowledgments
Lecture 1. Knots and Links, Reidemeister Moves
1.1. Main definitions
1.2. Reidemeister moves
1.3. Torus knots
1.4. Invertibility and chirality
1.5. Exercises
Lecture 2. The Conway Polynomial
2.1. Axiomatic definition
2.2. Calculations
2.3. Uniqueness and existence of the Conway polynomial
2.4. Chirality, orientation-reversal, and multiplicativity of the Conway polynomial
2.5. Exercises
Lecture 3. The Arithmetic of Knots
3.1. Boxed knots and their connected sum
3.2. The semigroup of boxed knots
3.3. Ordinary knots vs. boxed knots
3.4. Decomposition into prime knots
3.5. Some remarks about unknotting
3.6. Exercises
Lecture 4. Some Simple Knot Invariants
4.1. Stick number
4.2. Crossing number
4.3. Unknotting number
4.4. Tricolorability
4.5. Digression about orientable surfaces
4.6. Seifert surface of a knot
4.7. The genus of a knot
4.8. Exercises
Lecture 5. The Kauffman Bracket
5.1. Digression: statistical models in physics
5.2. The “state” of a (nonoriented) knot diagram
5.3. Definition and properties of the Kauffman bracket
5.4. Is the Kauffman bracket invariant?
5.5. Exercises
Lecture 6. The Jones Polynomial
6.1. Definition via the Kauffman bracket
6.2. Main properties of 𝐽(mskip 2𝑚𝑢⋅mskip 2𝑚𝑢)
6.3. Axioms for the Jones polynomial
6.4. Multiplicativity
6.5. Chirality and reversibility
6.6. Is the Jones polynomial a complete invariant?
6.7. Is 𝑉 a Laurent polynomial in 𝑞?
6.8. Knot tables revisited
6.9. Exercises
Lecture 7. Braids
7.1. Geometric braids
7.2. The geometric braid group 𝐵_{𝑛}
7.3. Digression on group presentations
7.4. Artin presentation of the braid group
7.5. Digression on undecidable problems
7.6. Closure of a braid
7.7. Exercises
Lecture 8. Discriminants and Finite Type Invariants
8.1. Discriminant of quadratic equations and real roots
8.2. Degree of a point w.r.t. a curve
8.3. Inertia index of a quadratic form
8.4. Gauss linking number
8.5. Exercises
Lecture 9. Vassiliev Invariants
9.1. Basic definitions
9.2. The one-term and four-term relations
9.3. Dimensions of the spaces 𝑉_{𝑛}
9.4. Chord diagrams
9.5. Vassiliev invariants of small order
9.6. Exercises
Lecture 10. Combinatorial Description of Vassiliev Invariants
10.1. Digression: graded algebras
10.2. The graded algebra of chord diagrams
10.3. The Vassiliev–Kontsevich theorem
10.4. Vassiliev invariants vs. other invariants
10.5. Exercises
Lecture 11. The Kontsevich Integrals
11.1. The original Kontsevich integral of a trefoil knot
11.2. Calculation of the integral for 𝑚=2
11.3. Kontsevich integral of the hump
11.4. Results
11.5. Exercises
Lecture 12. Other Important Topics
12.1. Knot polynomials
12.2. Virtual knots
12.3. Knots in 3-manifolds
12.4. Khovanov homology
12.5. Knot energy
12.6. Connections with other fields
Lecture 13. A Brief History of Knot Theory
13.1. Carl Friedrich Gauss: pictures of knots and the linking number
13.2. William Thompson, P.G. Tait, J.C. Maxwell, and knots as models of atoms
13.3. Henri Poincaré: surgery along the trefoil and the fundamental group
13.4. Max Dehn, Kurt Reidemeister, the German school, and the beginnings of knot theory
13.5. James Alexander, John Conway, their polynomial, and the skein relation
13.6. Vaughan Jones, Louis Kauffman, and the discoverers of the HOMFLY polynomial
13.7. Edward Witten, Michael Atiyah, and quantum field theory
13.8. Oleg Viro, Nikolay Reshetikhin, Vladimir Turaev, and a rigorous theory of links in manifolds
13.9. Wolfgang Haken, Friedhelm Waldhausen, Sergei Matveev, and the classification of knots
13.10. Victor Vassiliev and Mikhail Goussarov, and finite type invariants
13.11. Maxim Kontsevich, Dror Bar-Natan, Joan Birman, and the combinatorial theory of finite type invariants
13.12. Concluding remarks
Bibliography
Index
Series Titles
Back Cover

Citation preview

Knots, Links and Their Invariants Knots, Links and Their Invariants

Other important features of the book are the large number of original illustrations, numerous exercises and the absence of any references in the first eleven lectures. The last two lectures differ from the first eleven: they comprise a sketch of non-elementary topics and a brief history of the subject, including many references.

STML 101

This book is an elementary introduction to knot theory. Unlike many other books on knot theory, this book has practically no prerequisites; it requires only basic plane and spatial Euclidean geometry but no knowledge of topology or group theory. It contains the first elementary proof of the existence of the Alexander polynomial of a knot or a link based on the Conway axioms, particularly the Conway skein relation. The book also contains an elementary exposition of the Jones polynomial, HOMFLY polynomial and Vassiliev knot invariants constructed using the Kontsevich integral. Additionally, there is a lecture introducing the braid group and shows its connection with knots and links.

STUDENT MATHEMATICAL LIBRARY Volume 101

An Elementary Course in Contemporary Knot Theory A. B. Sossinsky

Sossinsky

For additional information and updates on this book, visit www.ams.org/bookpages/stml-101

AMS

STML/101

4-Color Process

148 pages spine: 5/16" finish size: 5.5” X 8.5” 50 lb stock

Knots, Links and Their Invariants An Elementary Course in Contemporary Knot Theory

STUDENT MATHEMATICAL LIBRARY Volume 101

Knots, Links and Their Invariants An Elementary Course in Contemporary Knot Theory A. B. Sossinsky

EDITORIAL COMMITTEE John McCleary Rosa C. Orellana (Chair)

Paul Pollack Kavita Ramanan

2020 Mathematics Subject Classification. Primary 55-xx, 51-xx, 20-xx. For additional information and updates on this book, visit www.ams.org/bookpages/stml-101 Library of Congress Cataloging-in-Publication Data Names: Sosinski˘ı, A. B. (Alekse˘ı Bronislavovich), author. Title: Knots, links and their invariants: an elementary course in contemporary knot theory / A. B. Sossinsky. Description: Providence, Rhode Island: American Mathematical Society, [2023] | Series: Student Mathematical Library, 1520-9121; volume 101 | Includes bibliographical references and index. Identifiers: LCCN 2022051584 |(paperback) ISBN 9781470471514 | (ebook) ISBN 9781470473112 Subjects: LCSH: Knot theory–Textbooks. | Link theory–Textbooks. | AMS: Algebraic topology. | Geometry. | Group theory and generalizations. Classification: LCC QA612.2 .S665 2023 | DDC 514/.2242–dc23/eng20230117 LC record available at https://lccn.loc.gov/2022051584 DOI: https://doi.org/10.1090/stml/101 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission@ ams.org.

c 2023 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

28 27 26 25 24 23

Contents

Foreword

xi

Permissions & Acknowledgments

xv

Lecture 1.

Knots and Links, Reidemeister Moves

1

§1.1. Main definitions

1

§1.2. Reidemeister moves

4

§1.3. Torus knots

8

§1.4. Invertibility and chirality

8

§1.5. Exercises

9

Lecture 2.

The Conway Polynomial

11

§2.1. Axiomatic definition

12

§2.2. Calculations

12

§2.3. Uniqueness and existence of the Conway polynomial

14

§2.4. Chirality, orientation-reversal, and multiplicativity of the Conway polynomial 19 §2.5. Exercises Lecture 3.

19

The Arithmetic of Knots

21

§3.1. Boxed knots and their connected sum

21

§3.2. The semigroup of boxed knots

22 v

vi

Contents §3.3. Ordinary knots vs. boxed knots

24

§3.4. Decomposition into prime knots

25

§3.5. Some remarks about unknotting

26

§3.6. Exercises

26

Lecture 4.

Some Simple Knot Invariants

29

§4.1. Stick number

29

§4.2. Crossing number

30

§4.3. Unknotting number

31

§4.4. Tricolorability

32

§4.5. Digression about orientable surfaces

33

§4.6. Seifert surface of a knot

35

§4.7. The genus of a knot

36

§4.8. Exercises

36

Lecture 5.

The Kauffman Bracket

39

§5.1. Digression: statistical models in physics

39

§5.2. The “state” of a (nonoriented) knot diagram

41

§5.3. Definition and properties of the Kauffman bracket

42

§5.4. Is the Kauffman bracket invariant?

43

§5.5. Exercises

46

Lecture 6.

The Jones Polynomial

47

§6.1. Definition via the Kauffman bracket

47

§6.2. Main properties of 𝐽( ⋅ )

49

§6.3. Axioms for the Jones polynomial

50

§6.4. Multiplicativity

51

§6.5. Chirality and reversibility

52

§6.6. Is the Jones polynomial a complete invariant?

52

§6.7. Is 𝑉 a Laurent polynomial in 𝑞?

53

§6.8. Knot tables revisited

54

§6.9. Exercises

55

Lecture 7.

Braids

57

Contents

vii

§7.1. Geometric braids

58

§7.2. The geometric braid group 𝐵𝑛

59

§7.3. Digression on group presentations

60

§7.4. Artin presentation of the braid group

62

§7.5. Digression on undecidable problems

63

§7.6. Closure of a braid

64

§7.7. Exercises

66

Lecture 8.

Discriminants and Finite Type Invariants

69

§8.1. Discriminant of quadratic equations and real roots

69

§8.2. Degree of a point w.r.t. a curve

71

§8.3. Inertia index of a quadratic form

72

§8.4. Gauss linking number

74

§8.5. Exercises

76

Lecture 9.

Vassiliev Invariants

77

§9.1. Basic definitions

78

§9.2. The one-term and four-term relations

80

§9.3. Dimensions of the spaces 𝑉𝑛

81

§9.4. Chord diagrams

81

§9.5. Vassiliev invariants of small order

83

§9.6. Exercises

84

Lecture 10.

Combinatorial Description of Vassiliev Invariants

87

§10.1. Digression: graded algebras

87

§10.2. The graded algebra of chord diagrams

88

§10.3. The Vassiliev–Kontsevich theorem

91

§10.4. Vassiliev invariants vs. other invariants

92

§10.5. Exercises

92

Lecture 11.

The Kontsevich Integrals

95

§11.1. The original Kontsevich integral of a trefoil knot

96

§11.2. Calculation of the integral for 𝑚 = 2

97

§11.3. Kontsevich integral of the hump

99

viii

Contents

§11.4. Results

100

§11.5. Exercises

101

Lecture 12.

Other Important Topics

103

§12.1. Knot polynomials

103

§12.2. Virtual knots

105

§12.3. Knots in 3-manifolds

106

§12.4. Khovanov homology

107

§12.5. Knot energy

108

§12.6. Connections with other fields

108

Lecture 13.

A Brief History of Knot Theory

111

§13.1. Carl Friedrich Gauss: pictures of knots and the linking number 112 §13.2. William Thompson, P.G. Tait, J.C. Maxwell, and knots as models of atoms 113 §13.3. Henri Poincaré: surgery along the trefoil and the fundamental group

114

§13.4. Max Dehn, Kurt Reidemeister, the German school, and the beginnings of knot theory 115 §13.5. James Alexander, John Conway, their polynomial, and 116 the skein relation §13.6. Vaughan Jones, Louis Kauffman, and the discoverers of 117 the HOMFLY polynomial §13.7. Edward Witten, Michael Atiyah, and quantum field theory

119

§13.8. Oleg Viro, Nikolay Reshetikhin, Vladimir Turaev, and a rigorous theory of links in manifolds 120 §13.9. Wolfgang Haken, Friedhelm Waldhausen, Sergei Matveev, and the classification of knots

120

§13.10. Victor Vassiliev and Mikhail Goussarov, and finite type invariants 121 §13.11. Maxim Kontsevich, Dror Bar-Natan, Joan Birman, and the combinatorial theory of finite type invariants 122 §13.12. Concluding remarks

123

Contents

ix

Bibliography

125

Index

127

Foreword

The present book consists of lecture notes for a one-semester introductory course in knot theory, but can also be used as a first textbook on the subject. The book differs from other textbooks and monographs on knot theory in that it presupposes very little knowledge of the traditional prerequisites for the course. Only a few basic facts of elementary Euclidean geometry, of 2-dimensional and 3-dimensional topology, and of group theory are required. We do not use such notions from topology as the fundamental group, homology theory, coverings, properties of homeomorphisms of ℝ3 . From group theory, we need only the basic definitions (group, homomorphism, subgroup, quotient group). As the result, the book does not contain such traditional topics of knot theory as the Wirtinger presentation of the fundamental group of knot complements, J.W. Alexander’s original definition of his polynomial invariant (via homomorphisms of 1-homology), Jones’s definition of his polynomial (involving representation theory, operator algebras and Markov’s theorem on the closure of braids), Victor Vassiliev’s original definition of finite type invariants (based on a cohomology spectral sequence of a filtered infinite-dimensional linear space). But it does contain a rigorous exposition of the most important results of knot theory obtained in the last forty years: the Alexander–Conway knot polynomial (defined on the basis of the Conway skein relation),

xi

xii

Foreword

the Jones polynomial (defined via the Kauffman bracket) and other knot polynomials (also obtained without the use of any advanced mathematics), the Vassiliev invariants (defined by axioms whose consistency is proved via the Kontsevich integrals). The set of invariants presented in this course is more powerful and aesthetically more satisfying that those coming from the fundamental group of knot complements, so that I only mildly regret the absence of Wirtinger’s algorithm in this course. The rigorous simplified exposition of knot theory presented in the course was made possible by the work of several researchers: by Louis Kauffman (his elementary construction of the bracket named after him led to a simple definition of the Jones polynomial), by John Conway and the author of these lectures (leading to the elementary definition of the Alexander polynomial given in this course), by Joan Birman, Maxim Kontsevich, Dror Bar Natan, Pierre Vogel, Sergey Duzhin, and others (leading to the axiomatization of Vassiliev’s theory of finite-type knot invariants and to a simpler proof of their existence). The final step in making this book elementary was made by using the elementary (although long, detailed, and delicate) proof of the existence of the Conway polynomial due to Roman Garaev. Another unusual aspect of the present course is the absence, in the first eleven lectures, of references to any books or articles, except to the books by D. Rolfsen [CD] and by S. Duzhin et. al., [DR], which are mostly mentioned to refer to various tables that they contain. To make the exposition really self contained, the course includes some preliminary material (outside of knot theory) that does not appear in the (truly very minimal!) list of prerequisites. These excursions are few and I call them “digressions”. They deal with (i) classification of surfaces and the Euler characteristic; (ii) graded algebras; (iii) algorithmic problems in algebra and topology. Lectures 12 (“Other Important Topics”) and 13 (“Brief History of Knot Theory”) were planned, but were not actually delivered to the students — semesters at the Independent University are thirteen weeks long and the midterm and final exam left no time for more than eleven lectures. Hence, the additional lectures are brief surveys rather than detailed expositions, they require mathematical knowledge beyond the declared prerequisites, and they contain numerous references.

Foreword

xiii

To conclude my comments about the contents of the course, let me stress the importance of the exercises, which appear at the end of each lecture. In order to learn knot theory (and most other branches of mathematics), it is more important to be able to solve problems than to memorize the theory. This is especially important when the book is used for individual study (say in a reading course); the reader should always try to solve a good part of the exercises after reading each chapter. If it turns out that such attempts are mostly unsuccessful, the reader should return to the material of the lecture, reread it, and try to figure out which parts of the material can be used to solve the elusive exercises. ∗

∗

∗

This course was given online in the framework of the Math in Moscow program in the fall semester of 2020 at the Independent University of Moscow. The course consisted of an hour-and-a-half lecture via Zoom and exercise classes of the same length each week. Students were of different nationalities and worked at home in Beijing, Singapore, London, Moscow, and San Francisco. All the students turned out to be unusually bright and motivated; almost all of them submited (by e-mail) correct solutions of practically all the problems. I am grateful to them for pointing out misprints and other errors in my original handouts (or should I say “e-mail sendouts”?) and complaining when the exposition was not clear enough. Because of the time differences, only part of the students participated in the lecture zoom sessions (the others would look at videos of the lectures on YouTube at a later hour), and there were two exercise classes. I conducted one, the other was done by my colleague Vladimir Medvedev, to whom I am grateful for helping to find original solutions to the exercises and pointing out some mistakes in the handouts. I am grateful to Yuri Thorkhov of MCNMO Publishers for the suggestion to publish of a small printrun of the lecture notes of the original course sent to the students, to Sergei Gelfand of the AMS for proposing the publication of a book based on a seriously revised version of the lecture notes, to Sergei Lvovsky for his highly qualified editorial work, and am especially grateful to Victor Shuvalov for authoring the illustrations, reformatting the text, and correcting a few errors.

Permissions & Acknowledgments

The American Mathematical Society gratefully thanks the following people and institutions for permission to reproduce figures and extracts. Image of Carl Friedrich Gauss, p. 112 Courtesy of Siegfried Detlev Bendixen. Published 1828 in “Astronomische Nachrichten”. Public Domain: https://commons.wikimedia.org/w/index.php?curid=2404149 Image of Henri Poincaré , p. 114 Courtesy of Ch. Wittmann—Henri Poincaré, available freely at Project Gutenberg. Public Domain: https://commons.wikimedia.org/w/index.php?curid=11437860 Image of Max Dehn, p. 115 Courtesy of Konrad Jacobs—Oberwolfach Photo Collection: https://opc.mfo.de/detail?photo_id=13906, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=86274458 Image of Kurt Reidmeister, p. 115 Courtesy of Konrad Jacobs—Oberwolfach Photo Collection: https://opc.mfo.de/detail?photo_id=13906, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=86274458

xv

xvi

Permissions & Acknowledgments

Image of John Conway, p. 116 Courtesy of “Thane Plambeck” https://www.flickr.com/photos/thane/20366806/ CC BY 2.0, https://commons.wikimedia.org/w/index.php ?curid=1781865 Image of Vaughan Jones, p. 117 Courtesy of David Monniaux—own work. CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1856032 Image of Louis Kauffman, p. 117 Courtesy of Louis H. Kauffman. Image of Edward Witten, p. 119 Courtesy Ojan—own work, Public Domain: https://commons.wikimedia.org/w/index.php?curid=3962763 Image of Michael Atiyah, p. 119 Courtesy of Gert-Martin Greuel—MFO: https://opc.mfo.de/detail?photoID=10118, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=3900577 Image of Oleg Viro, p. 120 Courtesy of Breithaupt, Katrin. https://opc.mfo.de/detail?photo_id=10655, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=6096801 Image of Nikolay Reshetikhin, p. 120 Courtesy of Lamprepair77—own work. Public Domain: https://commons.wikimedia.org/w/index.php?curid=12338637 Image of Vladimir Turaev, p. 120 Courtesy of Vladimir Turaev. Image of Wolfgang Haken, p. 121 Courtesy of Aehaken—own work. CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=79436419 Image of Sergei Matveev, p. 121 Courtesy of Sergei Matveev. Image of Victor Vassiliev, p. 122 Courtesy of Victor Vassiliev. Image of Mikhail Goussarov, p. 122 Courtesy of Michael Polyak.

Permissions & Acknowledgments Image of Maxim Kontsevich, p. 123 Courtesy of Maxim Kontsevich. Image of Joan Birman, p. 123 Courtesy of Joan Birman. Image of Dror Bar-Natan, p. 123 Courtesy of George M. Bergman. CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=6093580

xvii

Lecture 1

Knots and Links, Reidemeister Moves

In this lecture, we shall introduce knots and links, the main protagonists of this course. Intuitively, you can think of a knot as a string in 3-dimensional space whose extremities have been identified, of a link, as several such strings; these strings can be deformed, i.e., moved about in space, stretched and compressed, but they cannot be cut or glued. Two knots (or links) are considered equivalent if one can be deformed so as to have the same shape as the other (i.e., be isometric to the other). There are several ways to define knots, links, and the corresponding equivalence relation. In the first part of this course, we will use elementary geometric definitions of these notions — they appear in the next section. The main goal of the present lecture is to give some examples of knots and links, introduce the notions of knot diagram, link diagram, and their Reidemeister moves, and prove the Reidemeister Theorem that transforms 3-dimensional topological knot theory into a branch of 2dimensional combinatorial geometry.

1.1. Main definitions A (nonoriented) knot is defined as a closed broken line without self-intersections in Euclidean space ℝ3 . A (nonoriented) link is a set of pairwise 1

2

1. Knots and Links, Reidemeister Moves

disjoint closed broken lines without self-intersections in ℝ3 ; the set may be empty or consist of one element, so that a knot is a particular case of a link. An oriented knot or link is a knot or link supplied with an orientation of its line(s), i.e., with a chosen direction for going around its line(s). In what follows, we will be mostly dealing with oriented knots, but will explicitly indicate the orientation by arrows only if it is of importance in the case under consideration. Some well known knots and links are shown in the figure below. (The lines representing the knots and links appear to be smooth curves rather than broken lines; this is because the edges of the broken lines and the angles between successive edges are tiny and indistinguishable by the human eye :-)). The knot (a) is the right trefoil, (b), the left trefoil (it is the mirror image of (a)), (c) is the eight knot, (d) is the granny knot; the link (e) is called the Hopf link, (f) is the Whitehead link, and (g) is known as the Borromeo rings.

(a)

(e)

(b)

(c)

(f)

(d)

(g)

Figure 1.1. Examples of knots and links

Two knots (or links) 𝐾, 𝐾 ′ are called equivalent if there exists a finite sequence of Δ-moves taking 𝐾 to 𝐾 ′ , a Δ-move being one of the transformations shown in Figure 1.2; note that such a transformation may be performed only if triangle 𝐴𝐵𝐶 does not intersect any other part of the line(s). Figure 1.3 shows how a knot’s shape can be transformed by a succession of Δ-moves.

1.1. Main definitions

3

A

A

A

A B

C

B

B

C

C

C

Figure 1.2. Δ-moves

K

K 12

3 4

5 8

7 6

Figure 1.3. Modifying a knot by Δ-moves

As explained above, you can think of a knot as a thin elastic string in 3-dimensional space that can be deformed (i.e., stretched, compressed, and moved about), and consider two knots to be equivalent if one of them can be deformed to exactly the same shape as the other’s (i.e., made to be isometric to it). Using the term knot (or link), we often use this term to stand for the entire equivalence class of knots containing the given concrete knot. When two knots are in the same equivalence class, we often say that they have the same knot type. Thus when we say “the knot shown in Figure 1.3 is the left trefoil”, we mean that it is equivalent to the trefoil (say the one shown in Figure 1.1(a)). One of the basic problems of knot theory, often called the knot classification problem, is to determine whether any two given concrete knots are equivalent. An important particular case of that problem is the unknotting problem: given any concrete knot, determine if it is the unknot (also called trivial knot), i.e., the round circle. In this course, we shall be particularly interested in these two problems. Remark 1.1. In the examples in Figure 1.1, there are two trefoil knots — the “right” and the “left” one, (a) and (b). They are obviously mirror symmetric, but are not equivalent (the proof, which uses a powerful invariant, the Jones polynomial, will be given later in the course). You will find out, when you do Exercise 1.3, that the “eight knot” (Figure 1.1(c))

4

1. Knots and Links, Reidemeister Moves

is, on the contrary, equivalent to its mirror image. Later in the course, we will discuss this and similar facts (the “chirality” of knots), as well as the “invertibilty” of knots (whether their equivalence class changes or doesn’t change when their orientation is reversed). Remark 1.2. Most books on knot theory give a definition of knots equivalence different from ours. Namely, they define knots as closed curves in ℝ3 satisfying a local “tameness” condition and say that two knots 𝐾 and 𝐾 ′ are ambient isotopic if there exists an orientation-preserving homeomorphism ℎ ∶ ℝ3 → ℝ3 such that ℎ(𝐾) = 𝐾 ′ . This definition is equivalent to ours in the sense that the two definitions yield the same equivalence classes of knots and links. The proof of this fact is quite difficult, and we omit it. From now on, we will use the following terminology: instead of saying that two knots (links) are equivalent, we shall say they are isotopic, omitting the adjective “ambient” for brevity. Remark 1.3. This remark is for those readers who are familiar with the notions of homotopy and of isotopy, and it is purely terminological — it explains why the adjective “ambient” appears in the definition mentioned in Remark 1.2, so that the present remark can be skipped by the other readers. The point here is that if we declare two knots 𝐾 and 𝐾 ′ to be “equivalent” when there is an isotopy taking 𝐾 to 𝐾 ′ , then it is easy to show that all knots are equivalent to the unknot — see Figure 1.4.

1.2. Reidemeister moves All the knots and links in Figures 1.1 and 1.3 are pictured as knot diagrams or link diagrams, i.e., as projections in general position of the knot or link on the horizontal plane showing, at each double point, which of its branches passes above the other one; “in general position” means that no vertex is projected to another vertex, there is only a finite number of double points, and they are all transversal self-intersections. Thus the

Figure 1.4. All knots are (non-ambient) isotopic to the unknot

1.2. Reidemeister moves

5

double points become crossing points, at which one branch is an overpass, the other is an underpass. The projection (with double points instead of crossing points) is called the shadow of the link or knot diagram. The advantage of picturing knots and links as being (almost!) planar is that we pass from a 3-dimensional problem (which is not easy to visualize) to a planar one, which is easier to work with, especially after the so-called Reidemeister moves Ω1 , Ω2 , Ω3 are introduced: they are defined as shown in Figure 1.5.

↔

↔

Ω1

↔

↔

Ω2

Ω3 Figure 1.5. Reidemeister moves

These figures should be understood as follows. The pictures show only the part of the link located inside the disk bounded by a dashed line; each move does not modify the part of the link lying outside of the disk, but changes the part of the link that lies inside the disk as shown in the picture. Thus Ω1 creates/kills a little loop, Ω2 creates/kills a double overpass, Ω3 shifts a branch of the knot over a crossing point. Note that Ω3 in Fig. 1.5 can also be regarded as shifting a branch under a crossing point. We shall also need the following definition: two knot diagrams are called planar isotopic if their shadows can be obtained from each other by a finite sequence of moves shown in Figure 1.6; these moves are denoted by Pl1 and Pl2 . Reidemeister Theorem 1.1. Two link diagrams, in particular knot diagrams, 𝐿 and 𝐿′ , are isotopic if and only if 𝐿 can be taken to 𝐿′ by a finite sequence of Reidemeister moves and planar isotopy moves.

6

1. Knots and Links, Reidemeister Moves

Figure 1.6. Planar isotopy moves

Proof. The “if” part of the statement is obvious. The proof of the “only if” part is basically a general position argument — it suffices to show that any Δ-move can be replaced by Reidemeister moves. Suppose we are given a Δ-move [𝐴𝐵] ↦ [𝐴𝐶] ∪ [𝐵𝐶] (Figure 1.7). Without loss of generality, we can assume that the shadows of the edges [𝐴𝐷] or [𝐵𝐸] of our link issuing from 𝐴 and 𝐵 do not go inside triangle 𝐴𝐵𝐶. Indeed, if (say) [𝐴𝐷] goes into the triangle, we choose a point 𝐴′ on 𝐴𝐶 near 𝐴 and perform an Ω1 move as shown in Figure 1.7, obtaining the new triangle 𝐴′ 𝐵𝐶 such that the edge issuing from 𝐴′ does not go into the new modified triangle. C

C

D B A

A

D B

A Figure 1.7. Modifying triangle 𝐴𝐵𝐶

Now let us note that the branches of our link whose shadows intersect triangle 𝐴𝐵𝐶 either lie entirely above it or entirely below it (otherwise these branches would pierce the triangle, which is forbidden by the definition of Δ-moves). Let us partition triangle 𝐴𝐵𝐶 into small triangles 𝑋𝑌 𝑍 of different types with a designated side 𝑋𝑌 ; we will perform successive Δ-moves using these triangles, namely the Δ-moves that take [𝑋, 𝑌 ] to[𝑋, 𝑍] ∪ [𝑍, 𝑌 ] or vice versa.

1.2. Reidemeister moves

7

Triangles of type I contain only two segments of the shadow of 𝐿, and these segments intersect inside triangle 𝐴𝐵𝐶. Triangles of type II contain one vertex of the shadow, and the lines issuing from that vertex either both intersect [𝑋𝑌 ] or both intersect [𝑋, 𝑍] ∪ [𝑍, 𝑌 ]; triangles that contain only one segment of the shadow, and that segment intersects both [𝑋, 𝑍] and [𝑍, 𝑌 ], are also of type II. Triangles of type III contain only one segment of the shadow, and that segment intersects both [𝑋, 𝑌 ] and [𝑋, 𝑍] ∪ [𝑍, 𝑌 ]. Triangles of type IV are empty: they contain no vertices and no parts of lines of the shadow. Such a partition can be constructed as follows. First, for each crossing and each vertex of the shadow of 𝐿, we construct nonintersecting little triangles of type I and II containing the vertex or crossing point, respectively, and then partition the remaining part of triangle 𝐴𝐵𝐶 into triangles of type III and IV. Now, instead of the given Δ-move, we progressively move from [𝐴𝐵] to [𝐴𝐶]∪[𝐶𝐵], performing Reidemeister moves associated to each of the little triangles. Namely, for each triangle of type I we do an Ω3 move, for each triangle of type II, an Ω2 move, for each triangle of type III, a Pl2 move, finally for each triangle of type IV, a Pl1 move. This concludes the proof. □ Remark 1.4. The fundamental importance of the theorem is due to the fact that it reduces a difficult 3-dimensional topological problem to Z

Z

I

X

Y

IV

X

Y

Z

X

Y

Z

II

X

Z

III

Z

II

Y

X

II

Y

X

Figure 1.8. Triangles of types I, II, III, and IV

Y

8

1. Knots and Links, Reidemeister Moves

a (simpler) problem in 2-dimensional geometric combinatorics. Actually, some textbooks in knot theory define links as link diagrams up to Reidemeister moves, thus transforming 3-dimensional knot theory into a branch of 2-dimensional geometric combinatorics, thereby hiding its 3-dimensional nature. In this course, we prefer to stay in three dimensions.

1.3. Torus knots If 𝑝, 𝑞 are coprime positive integers, the torus knot 𝑇(𝑝, 𝑞) is defined as the closed curve lying on the standard torus and winding 𝑝 times around the meridian of the torus and 𝑞 times around its parallel. Figure 1.9 shows two torus knots. The trefoil is obviously a torus knot, namely 𝑇(3, 2). The mirror image of a torus knot is also a torus knot, not isotopic to the given one. Torus knots are classified (up to mirror symmetry) by pairs of coprime natural numbers.

Figure 1.9. The torus knots 𝑇(5, 2) and 𝑇(4, 3)

One can also study torus links — their definition is the object of Exercise 1.8.

1.4. Invertibility and chirality For an orientable knot 𝐾, let us denote by 𝐾⃖ the knot obtained by reversing the orientation of 𝐾 and by 𝐾 ∗ , the mirror image of 𝐾. A knot 𝐾

1.5. Exercises

9

is called invertible if 𝐾⃖ = 𝐾, plus-amphicheiral if 𝐾 ∗ = 𝐾, and minusamphicheiral if 𝐾 ∗ = 𝐾.⃖ In what follows, we use Rolfsen’s notation for knots (e.g. 31 for the left trefoil and 41 for the eight knot). The reader can look at the other knots indicated below by googling “Rolfsen knot tables”. The following five logically possible combinations of invertibility and chirality are actually realized by specific knots: (1) all four knots 𝐾, 𝐾,⃖ 𝐾 ∗ , 𝐾⃖ ∗ are the same — by the eight knot; (2) all four knots 𝐾, 𝐾,⃖ 𝐾 ∗ , 𝐾⃖ ∗ are different — by the knot 932 ; (3) the knot 𝐾 is invertible and differs from its mirror image — by the trefoil knot 31 ; (4) the knot 𝐾 is noninvertible and plus-amphicheiral — by the knot 12427 ; (5) the knot 𝐾 is noninvertible and minus-amphicheiral — by the knot 8 17 . We will be able to prove these facts only much later in the course by using various knot invariants.

1.5. Exercises 1.1. Using Reidemeister moves, show that the two diagrams of the unknot with opposite orientations are equivalent. ? K

K

1.2. Using Reidemeister moves, show that the two knot diagrams 𝐾1 and 𝐾2 represent the same knot.

?

K1

K2

10

1. Knots and Links, Reidemeister Moves

1.3. Using Reidemeister moves, show that reversing the orientation transforms the knot diagram of the right trefoil into an isotopic knot. 1.4. Which of these knots represents the right trefoil? The figure eight knot? The unknot?

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

1.5. Using Reidemeister moves, show that reversing the orientation transforms the knot diagram of the eight knot into a diagram of the same eight knot. 1.6. Find a knot diagram with five crossings which is a torus knot. What are the values of 𝑝 and 𝑞 for that knot? 1.7. Find a knot diagram with 17 crossings which is a torus knot. What are the values of 𝑝 and 𝑞 for that knot? 1.8. Give a reasonable definition of torus link and classify torus links up to mirror symmetry.

Lecture 2

The Conway Polynomial

In this lecture, we define the values of the Alexander–Conway polynomial, which is an isotopy invariant of knots and links, by means of three simple axioms due to John Conway. These axioms include the so-called Conway skein relation, an unusual geometric-combinatorics relation, which opened the way to the modern theory of knot invariants. We then learn to calculate the values of the Alexander–Conway polynomial for concrete knots and links and see how well that polynomial distinguishes nonisotopic knots and nonisotopic links. It turns out that the Alexander–Conway polynomial is a strong, but not a complete invariant. We conclude the lecture by giving an elementary proof of the fact that the Alexander–Conway polynomial actually exists, i.e., that there is a unique map ∇ of the set of (isotopy classes of) links to the ring ℤ[𝑥] of one-variable polynomials with integer coefficients such that ∇ satisfies the three axioms. This proof is based on recent work by R. Garaev. Conway proved that such a polynomial ∇𝐿 (𝑥) exists and is unique by showing that the three axioms are satisfied by the polynomial 𝐴𝐿 (𝑡) ∈ ℤ[𝑡, 𝑡−1 ] originally defined by J.W. Alexander if one changes the variable according to the rule 𝑥 ↔ √𝑡 − 1/√𝑡. Alexander defined his polynomial 𝐴𝐿 (𝑡) as an element of ℤ[𝑡, 𝑡−1 ] by a concrete rather sophisticated topological construction involving 1-homology groups and cyclic coverings.

11

12

2. The Conway Polynomial

2.1. Axiomatic definition If we want to prove that two knot (link) diagrams represent the same knot (link), it suffices to find a series of Δ-moves (or Reidemeister moves) taking one to the other. But what must we do to prove that two knot (link) diagrams represent different knots (links)? We must use an invariant. Here we shall use the Conway polynomial (Conway’s version of the Alexander polynomial), defined as follows. To each oriented link (in particular knot) diagram 𝐿, a polynomial with integer coefficients in the variable 𝑥, called the Conway polynomial of the link 𝐿 and denoted by ∇(𝐿) or ∇𝐿 (𝑥), is assigned; this assignment must satisfy to the three following conditions (Conway axioms): (I) Invariance: If 𝐿 is isotopic to 𝐿′ , then ∇(𝐿) = ∇(𝐿′ ). (II) Normalization: ∇(○) = 1 for the unknot ○. (III) Skein relation: ∇(𝐿+ ) − ∇(𝐿− ) = 𝑥 ⋅ ∇(𝐿∘ ) ∇(

) − ∇(

) = 𝑥∇(

).

The above relation should be understood as follows: we are given three links 𝐿+ , 𝐿− , 𝐿∘ that are identical outside the small disks bounded by the three dashed circles, inside which they are as shown in the picture, and their Conway polynomials satisfy the displayed relation. It turns out that these three axioms are quite sufficient for calculating the values of the Conway polynomial of concrete knots and links.

2.2. Calculations Let us calculate the Conway polynomial of the two-component trivial link ○○. Using (III), and then (I) and (II) twice, we obtain 𝑥∇(

) = ∇( 𝑥 ⋅ (?)

Thus ∇(○○) = 0.

=

) − ∇( 1

−

). 1

2.2. Calculations

13

Now let us calculate the Conway polynomial of the right Hopf link, i.e., the Hopf link with oppositely oriented circles. We have

∇(

) − ∇( ?

−

) = 𝑥∇( 0

=

). 𝑥⋅1

Thus ∇(right Hopf link) = 𝑥. Finally, let us calculate the Conway polynomial of the right trefoil. We have

∇(

) − ∇( ?

−

) = 𝑥∇( 1

=

). 𝑥⋅𝑥

Thus ∇(right trefoil) = 1 + 𝑥2 . What do those calculations show? They show that the Hopf link cannot be unlinked (i.e., is not isotopic to the trivial two-component link) and that the right trefoil cannot be unknotted (i.e., is not isotopic to the unknot) provided that we know that an assignment 𝐿 ↦ ∇(𝐿) satisfying axioms (I), (II), (III) exists and is unique. This will be proved in the next section, and now we shall continue similar calculations assuming that this has been proved. The next link whose Conway polynomial we will find is the left Hopf link (Exercise 2.1), then we calculate that of the left trefoil (Exercise 2.2), obtaining ∇(left trefoil) = 1+𝑥2 and then of the eight knot (Exercise 2.3), ∇(eight knot) = 1 − 𝑥2 . These calculations show that the eight knot is not a trefoil, nor is it the unknot, and that the Conway polynomial does not distinguish the right trefoil from the left one. Actually, the two trefoils are not isotopic to each other — we shall prove this by using the Jones polynomial (Lecture 6). Thus we see that the Conway polynomial is not a complete invariant.

14

2. The Conway Polynomial

2.3. Uniqueness and existence of the Conway polynomial We shall need the following lemma, which will also be useful in other contexts, in particular in Lecture 4. Trivialization Lemma 2.1. Any link (knot) diagram can be transformed into a trivial link (knot) diagram by an appropriate series of crossing changes. Proof. We shall prove the lemma for knots and leave the (straightforward) generalization to arbitrary links to the reader. Suppose we are given a knot diagram 𝐾. Let us choose an arbitrary point 𝑃 on 𝐾 and move along the knot in the direction of orientation until we reach the first crossing point 1; if 1 is an underpass for us, we do not make a crossing change at 1; if 1 is an overpass for us, we do make a crossing change at 1. We continue our motion until we come to a new crossing point 2 (it may happen that we will first come through 1 again if the knot has a little loop near 1, as in the Example in Figure 2.1(a)); if 2 is an underpass for us, we do not make a crossing change at 1, otherwise we do. We then go on to the third crossing point 3 and make (or do not make) a crossing change there according to the same rule as before, and continue further in the same way until we return to 𝑃. As the result, we obtain a new knot diagram 𝐾 ′ . For the example in Figure 2.1(a), it is shown in Fig. 2.1(b). Now let us prove that 𝐾 ′ is a diagram of the unknot. To do that we shall trace out a knot 𝐾 ″ in space (near the horizontal plane “almost containing” 𝐾 ′ ) that will be obviously equivalent to 𝐾 ′ . To do that, we start at 𝑃 and move along and vertically above the curve 𝐾 ′ , uniformly rising upward very very slowly and as we go around 𝐾 ′ until we come back to some point 𝑃 ′ near 𝑃, and then move down to close up the curve at 𝑃. To prove that 𝐾 ″ (and hence 𝐾 ′ !) is the unknot, it suffices to look at 𝐾 ″ from a point in the horizontal plane: we will then see an oscillating closed curve without self-intersections. This proves the lemma. □ Theorem 2.1. There exists a unique assignment ∇ ∶ ℒ → ℤ[𝑥] that takes any oriented link diagram 𝐿 ∈ ℒ to a polynomial in 𝑥 with integer coefficients so that the following three axioms hold: I. Invariance: ∇(𝐿) = ∇(𝐿′ ) if 𝐿 is (ambient) isotopic to 𝐿′ .

2.3. Uniqueness and existence of the Conway polynomial

4

4

5

5 3

1

15

3

1

2

P

2

P K

K (a)

K  (b) Figure 2.1. Trivializing a knot diagram by crossing changes

II. Normalization: ∇(○) = 1, where ○ denotes the unknot. III. Conway skein relation: ∇(𝐿+ ) − ∇(𝐿− ) = 𝑥 ∇(𝐿∘ ), or ∇(

) − ∇(

) = 𝑥∇(

)

where 𝐿+ , 𝐿− , 𝐿∘ are identical link diagrams outside a small disk, inside which they are as pictured inside the dashed circles. Proof. We will show by induction on 𝑛 that for any link diagram 𝐿 with ≤ 𝑛 crossings, the polynomial ∇(𝐿) is uniquely determined by the three axioms. The base of induction is satisfied, because a link diagram 𝐿 without crossings is either the unknot (in which case its polynomial is equal to 1 by axioms I and II) or a trivial 𝑚-component link with 𝑚 ≥ 2, in which case its polynomial is zero, as can be proved similarily to the case 𝑚 = 2 considered above). We assume (induction hypothesis) that for any link diagram 𝐿 with ≤ 𝑛 − 1 crossing points ∇(𝐿) is uniquely determined by the three axioms. Let us consider an arbitrary link diagram with 𝑛 crossings and

16

2. The Conway Polynomial

choose one of the crossings. Denote this link diagram by 𝐿+ if the chosen crossing is positive and by 𝐿− if it is negative. In either case, by the skein relation, we have (1)

∇(𝐿+ ) − ∇(𝐿− ) = 𝑥 ∇(𝐿∘ ),

where 𝐿∘ denotes the corresponding link with 𝑛 − 1 crossings. Since we know the right-hand side (by the induction hypothesis) it remains to show that one of the two terms on the left-hand side is uniquely determined by the axioms. To do that, we will show, for a fixed value of 𝑛, by induction on the unknotting number 𝑘 (i.e., the smallest 𝑘 such that the link can be trivialized by 𝑘 crossing changes, see the Trivialization Lemma) that, for any link diagram 𝐿 with 𝑛 crossings and unknotting number 𝑘 ≤ 𝑛, the polynomial ∇(𝐿) is uniquely determined by the axioms. When 𝑘 = 0, this is true because the assertion is the same as the base of induction (𝑛 = 0) considered above. Let us assume that it is true for 𝑘−1 and prove it for 𝑘. Consider a link diagram with 𝑛 crossings and unknotting number 𝑘. Then there is a crossing in the diagram at which the crossing change produces a link diagram with unknotting number 𝑛 − 1. If that crossing is positive, we denote our diagram by 𝐿+̃ , and by 𝐿−̃ if it is negative. Then, in any case, by the skein relation, we have (2)

∇(𝐿+̃ ) − ∇(𝐿−̃ ) = 𝑥 ∇(𝐿∘̃ ),

where 𝐿∘̃ is the corresponding link diagram with 𝑛 − 1 crossings. We know the value of the right-hand side (by the induction hypothesis for 𝑛) and so it suffices to show that one of the two terms on the left-hand side is determined by the axioms. But this follows from the induction hypothesis (for 𝑘 this time!), since one of these terms (not the one containing the chosen link diagram, but the other one!) must have unknotting number 𝑛 − 1 (because of our special choice of crossing). This concludes the induction on 𝑘 and therefore the induction on 𝑛, proving uniqueness. Let us pass to the proof of existence. First, we define the notion of ordered link diagram as an oriented link diagram whose components are ordered: 𝐿 = 𝐾1 , . . . , 𝐾𝑠 . We claim that to any nonordered oriented link diagram 𝐿, we can associate a finite sequence (𝑐−1 = 0, 𝑐 0 , 𝑐 1 , 𝑐 2 , . . .) of integer invariants so

2.3. Uniqueness and existence of the Conway polynomial

17

that (3)

∇𝐿 (𝑥) ≔ 𝑐 0 + 𝑐 1 𝑥 + 𝑐 2 𝑥2 , . . .

satisfies the Conway axioms, in particular the Conway skein relation and isotopy invariance. We need more definitions. The sign of a crossing of a (nonordered) oriented link diagram is +1 if the crossing is positive, −1 if the crossing is negative, and 0 if crossing at that point no longer occurs. If 𝑎 is a crossing of the link 𝐿, we denote by 𝜀(𝑎, 𝐿) the sign of the crossing 𝑎, by 𝐾𝑎 , the link obtained by changing the sign of 𝑎, and by 𝐾[𝑎] , the nonordered link diagram obtained by removing the crossing (replacing it by two “parallel” curves as in the right-hand side of Conway’s skein relation). It immeditely follows from (3) and the above definitions that the skein relation is satisfied. It remains to prove the isotopy invariance of ∇𝐿 (𝑥). We will need the following Lemma 2.2. Let (𝛼, 𝛽) be a pair of integer invariants such that 𝛽(𝑇) = 0 for any (nonordered) trivial link diagram 𝑇 having two components or more. Assume that these invariants satisfy the following relation 𝛽(𝐾) − 𝛽(𝐾𝑎 ) = 𝜀(𝑎, 𝐾) for any nonordered oriented link diagram 𝐾. Then there exists a polynomial invariant 𝛾 such that (i) 𝛾(𝐿) − 𝛾(𝐿𝑎 ) = 𝜀(𝑎, 𝐿)𝛾(𝐿𝑎 ) (ii) 𝛾(𝐿) = 0 for any (nonordered) trivial link diagram 𝐿 having two components or more. We claim that Lemma 2.2 implies the main theorem, Theorem 2.1. The proof of this fact is the subject of Exercise 2.11. We now state another lemma, very much like Lemma 2.2, but in which 𝛾 is an invariant of ordered links. Lemma 2.3. Let (𝛼, 𝛽) be a pair of integer invariants such that 𝛽(𝑇) = 0 for any ordered trivial link diagram 𝑇 having two components or more. Assume that these invariants satisfy the following relation 𝛽(𝐿) − 𝛽(𝐿𝑎 ) = 𝜀(𝑎, 𝐿) for any ordered oriented link diagram 𝐿. Then there exists a polynomial invariant 𝛾 such that (i) 𝛾(𝐿) − 𝛾(𝐿𝑎 ) = 𝜀(𝑎, 𝐿)𝛾(𝐿𝑎 ) (ii) 𝛾(𝐿) = 0 for any ordered trivial link diagram 𝐿 having two components or more.

18

2. The Conway Polynomial

We claim that Lemma 2.3 implies Lemma 2.2. The proof of this fact is the subject of Exercise 2.12. So it remains to prove Lemma 2.3, i.e., construct the invariant 𝛾 and show that it has the required properties. To construct 𝛾, we need some more definitions and notation. From now on, the word diagram will stand for an ordered diagram, a contour is a closed oriented broken line, a family of contours is an ordered collection of contours whose vertices are in general position. Suppose that 𝜆 = (𝜆1 , . . . , 𝜆𝑛 ) is a family of contours, 𝜉 = (𝜉1 , . . . , 𝜉𝑛 ) is an ordered collection of points such that 𝜉𝑖 ∈ 𝜆𝑖 , and 𝜉𝑖 is not a self-intersection point of 𝜆𝑖 for 𝑖 = 1, . . . , 𝑛; then the pair (𝜆, 𝜉) is called a family of contours with marked points. For a diagram 𝐿 with components 𝜆1 , . . . , 𝜆𝑛 , we will say that 𝜆𝑖 lies under 𝜆𝑗 if any edge of 𝜆𝑖 either does not intersect 𝜆𝑗 or if any edge of 𝜆𝑖 lies under an edge of 𝜆𝑗 . By Lemma 2.1, there exists a collection of points 𝜉1 , . . . , 𝜉𝑛 such that the family of collections consists of 𝑛 unknotted contours such that the contour 𝜆𝑠 lies under 𝜆𝑡 whenever 𝑠 < 𝑡. Let 𝑥1 , . . . , 𝑥𝑗 be some crossings (not necessarily different pairwise) of some diagram 𝐿 and let 𝑥 ≔ (𝑥1 , . . . , 𝑥𝑗 ). Then 𝐿𝑥 denotes the diagram obtained from 𝐿 by reversing all the crossings 𝑥1 , . . . , 𝑥𝑗 . Now we can construct the function 𝛾. To each family 𝜆 of contours we associate a family (𝜆, 𝜉(𝜆)) of contours with marked points in all possible ways. Denote by 𝔹 the set of all pairs (𝑥, 𝜆), where 𝜆 is a family of contours and 𝑥 is an ordered collection of certain crossings (not necessarily distinct) of the diagram (𝜆, 𝜉(𝜆)). If the collection 𝑥 is empty, then we write the pair (𝑥, 𝜆) as (𝜆). Let us define a function 𝛾 ̂ ∶ 𝔹 → 𝔹 by setting 𝛾(𝑥, ̂ 𝑎, 𝜆) ≔ 𝛾(𝑥, ̂ 𝜆) − 𝜀(𝑎, (𝜆, 𝜉(𝜆)))𝛽((𝜆, 𝜉(𝜆))𝑥,[𝑎] )

and

𝛾 ̂ ≔ 0.

For each diagram 𝐿 obtained from an ordered collection 𝜆 supplied with the additional information about pairs edges of contours lying above or below each other, we choose an ordered collection 𝑦 = 𝑦(𝐿, ((𝜆, 𝜉(𝜆)) of crossings such that ((𝜆, 𝜉(𝜆)) = 𝐿. Now we can define the function 𝛾 required in Lemma 2.3 by setting 𝛾(𝐿) ≔ 𝛾(𝑦, ̂ 𝜆).

2.5. Exercises

19

It is easy to see that 𝛾(𝐿) is well defined. Of course, this definition, involving a huge amount of new terms and new notation, is difficult to grasp. To really understand it, I suggest that the reader choose some concrete, but fairly simple, diagram 𝐿 and perform all the required constructions, drawing pictures at each step. In particular, the reader should understand the geometric reasons for the key next-to-last step: the choice of the ordered collection 𝑦 of crossings. To prove Theorem 2.1, it remains to show that 𝛾 is an isotopy invariant. This is the subject of Exercise 2.13. □

2.4. Chirality, orientation-reversal, and multiplicativity of the Conway polynomial Theorem 2.2. (a) The knot 𝐾⃖ obtained from an oriented knot 𝐾 by reversing its orientation has the same Conway polynomial: ∇(𝐾)⃖ = ∇(𝐾). (b) The mirror image 𝐾 ∗ of an oriented knot 𝐾 has the same Conway polynomial: ∇(𝐾 ∗ ) = ∇(𝐾). (c) The Conway polynomial is multiplicative with respect to the connected sum operation # defined in Lecture 3, i.e., ∇(𝐿1 # 𝐿2 ) = ∇(𝐿1 ) ⋅ ∇(𝐿2 ) for all oriented links 𝐿1 , 𝐿2 . The proofs of (a) and (b) are the object of Exercises 2.8 and 2.9, respectively. I know no elementary proof of (c) (multiplicativity) — the only proof in the literature involves the so-called HOMFLY polynomial (see Lecture 12).

2.5. Exercises 2.1. Compute the Conway polynomial of the left Hopf link. 2.2. Compute the Conway polynomial of the left trefoil. Is it the same as that of the right trefoil? Can you conclude that the right and left trefoils are isotopic knots?

20

2. The Conway Polynomial

2.3. Show that the figure eight knot is not isotopic to the trefoil by computing its Conway polynomial. 2.4. Is the granny knot isotopic to one of the trefoils? To the figure eight knot? To the unknot? 2.5. Using Reidemeister moves, show that reversing the orientation transforms the knot diagram of the right trefoil into an isotopic knot. 2.6. Show that the Conway polynomial of a link diagram consisting of two components located in different half planes is equal to zero. 2.7. Show that the Conway polynomial of the trivial 𝑚-component knot is zero when 𝑚 ≥ 2. 2.8. Show that the Conway polynomial of a knot 𝐾 does not change if we replace 𝐾 by its inverse knot 𝐾⃖ (i.e., the knot obtained from 𝐾 by reversing its orientation). 2.9. Show that the Conway polynomial of a knot 𝐾 does not change if we replace 𝐾 by its mirror image 𝐾 ∗ . 2.10. Does the analogue of Theorem 2.2 (a), (b) hold for links? 2.11. Prove that Lemma 2.2 implies Theorem 2.1. 2.12. Prove that Lemma 2.3 implies Lemma 2.2. 2.13. Prove that 𝛾 is an isotopy invariant.

Lecture 3

The Arithmetic of Knots

In this lecture, we introduce a composition operation (called connected sum) in the set of (isotopy equivalence classes of) knots and study the algebraic structure thereby obtained. In order to do that, we shall put knots into boxes.

3.1. Boxed knots and their connected sum By definition, a boxed knot is a non-self-intersecting polygonal curve in a cube (or a in rectangular parallelopipedon) joining the centers of two of its opposite faces. Boxed knots are oriented (from left to right in the figures). Two boxed knots are called equivalent or (ambient) isotopic if there exists a finite sequence of Δ-moves (see Section 1.2 in Lecture 1) performed inside the cube and taking one curve to the other. The composition or connected sum of two boxed knots 𝐾1 and 𝐾2 , denoted by 𝐾1 # 𝐾2 , is obtained by joining the boxes containing them as shown in Figure 3.1. Our immediate aim is to investigate the algebraic structure of the set 𝒦 (of equivalence classes of boxed knots) w.r.t. the connected sum operation # . It turns out that this structure is surprisingly similar to that of the set of natural numbers w.r.t. multiplication, with the unknot playing the role of the unit 1. 21

22

3. The Arithmetic of Knots

=

#

Figure 3.1. Connected sum of two knots

3.2. The semigroup of boxed knots The connected sum operation is obviously well defined on isotopy classes of boxed knots and possesses the two following important properties. I. The connected sum operation is associative and commutative: 𝐾1 # 𝐾2 = 𝐾2 # 𝐾1 ,

(𝐾1 # 𝐾2 ) # 𝐾3 = 𝐾1 # (𝐾2 # 𝐾3 ).

II. There are no inverse elements under the connected sum operation, i.e., 𝐾 # 𝐾 ′ = ○ ⟹ 𝐾 = 𝐾 ′ = ○. Associativity is obvious, the proof of commutativity is shown in Fig. 3.2 for a concrete example, but the construction is clearly general.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.2. Commutativity of the connected sum operation

3.2. The semigroup of boxed knots

23

The proof of property II is similar to the following “proof” of the “equality” 1 = 0, 1 = 1 + 0 + 0 + 0 + . . . = 1 + (1 − 1) + (1 − 1) + (1 − 1) + . . . = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + . . . = (1 − 1) + (1 − 1) + (1 − 1) + . . . = 0 + 0 + 0 + . . . = 0. The proof of property II is sketched in Figure 3.3.

K=

K

=

K

=

¯ K#K

¯ K#K

¯ K#K

¯ K#K

¯ K#K

...

...

=

=

Figure 3.3. Nonexistence of inverse knots

From the intuitive point of view, property II means that if you have a nontrivial knot tied at one end of a rope, then it is impossible to tie

24

3. The Arithmetic of Knots

another knot at the other end so that when you pull the two ends apart, the two knots will cancel each other. In particular, if you tie a right trefoil at one end of a rope and a left trefoil at the other end, you get a knot, which is not trivial by Exercise 4.11). Unlike the above “proof” of one equaling zero, the proof of assertion II can be made rigorous by using the definition of knot equivalence indicated in Remark 1.2 (Lecture 1), but we omit the details.

3.3. Ordinary knots vs. boxed knots Before we continue the study of the algebraic structure of (𝒦, # ), let us compare knots regarded as closed polygonal curves (i.e., knot theory as developed in Lecture 1) to boxed knots. Fortunately, the two notions turn out to yield the same theory — this follows from the next statement. Proposition 3.1. There is a canonical bijection between the equivalence classes of boxed knots and the isotopy equivalence classes of oriented knots (as defined in Section 1.1, Lecture 1). Sketch of the proof. To any concrete boxed map we assign an oriented knot by joining the endpoints of the boxed knot by a polygonal non-selfintersecting curve lying outside the box in the vertical plane containing the endpoints. It is obvious that this assignment is well defined on equivalence classes, and it is also obvious that it is surjective. Objectivity also seems obvious, but it isn’t — its rigorous proof involves some delicate results on the topology of ℝ3 and is omitted. □ The canonical bijection carries over to ordinary knots all the structures that we have considered for boxed knots, in particular the connected sum operation. The latter can be defined directly for ordinary knots — how this is done is shown in Figure 3.4.

K1

,

K2

⇒

K1 #K2

Figure 3.4. Connected sum of ordinary oriented knots

3.4. Decomposition into prime knots

25

Figure 3.5. A composite knot

The connected sum of two knots 𝐾1 and 𝐾2 is obtained by placing the two knots in different half spaces of ℝ3 and joining them by two strands as shown in Figure 3.4 (note that this is done so that the orientations are in accord). We have avoided that definition because it is rather difficult to prove that it does not depend on the choice of representatives in the equivalence classes of the given knots.

3.4. Decomposition into prime knots A nontrivial boxed knot is called prime if it cannot be presented as the sum of two nontrivial knots, i.e., 𝐾 = 𝐾1 # 𝐾2 ⟹ 𝐾1 = ○ or 𝐾2 = ○. If a knot is not prime, we say that it is composite. An example of a composite knot is shown in Figure 3.5. The reader is invited to decompose this knot into (three) prime knots (Exercise 3.1). The main result of this lecture is the following

26

3. The Arithmetic of Knots

Theorem 3.1. The set of isotopy classes of knots w.r.t. the connected sum operation is a commutative semigroup (𝒦, # ) without inverse elements and with unique (up to order) decomposition into prime knots, i.e., 𝐾 ∈ 𝒦, 𝐾 ≠ ○ ⟹ ∃! {𝑃1 , . . . , 𝑃𝑛 } ∶ 𝐾 = 𝑃1 # . . . # 𝑃𝑛 , where the 𝑃𝑖 are prime knots. The proof of the second part of this theorem (concerning prime knot decomposition) is not elementary and therefore lies outside the framework of this course. The existence of a finite prime sum decomposition of any knot is proved by using properties of the fundamental group of the complement of the knot. The proof of uniqueness (up to order) was obtained in 1948 by Horst Schubert, and it involves some delicate 3-dimensional topology.

3.5. Some remarks about unknotting When analyzing Reidemeister moves, we may notice that some of them decrease the number of crossing points of the given knot diagrams, and this leads to the idea that by using such Reidemeister moves, one can obtain a simple unknotting algorithm: apply such moves until all the crossing points disappear — then the knot is trivial, but if at some stage no such moves are possible, declare that the knot is nontrivial. However, this idea doesn’t work: there exist knot diagrams representing the trivial knot that require using Reidemeister moves that increase the number of crossing points. Such an example is shown in Figure 3.6. Its justification is the subject of Exercise 3.3. This example is the manifestation of a fundamental principle of combinatorics and of the philosophy of everyday life. In combinatorics, the principle says that “greedy algorithms” (which aim at solving problems by performing local simplifications at each step) don’t always work. In real life, in order to simplify a given problem situation, it is sometimes necessary to first make it more complicated, and only then can one solve the problem.

3.6. Exercises 3.1. Represent the composite knot in Figure 3.5 as the composition of prime knots placed in boxes.

3.6. Exercises

27

Figure 3.6. A clever unknot

3.2. Draw a knot diagram of a composite knot which is the sum of two prime knots each with 5 crossings or less so that it is not easy to identify those two prime knots. 3.3. Show, using Reidemeister moves, that the knot in Figure 3.6 is indeed the unknot, and verify that no Reidemeister moves that decrease the number of crossings can be applied to it. 3.4. Give an example of a knot (substantially different from the knot in Figure 3.6) which cannot be unknotted by Reidemeister moves that do not increase the number of crossing points. 3.5. We know that reversing the orientation of one component of the Hopf link changes the sign of its Conway polynomial. Is this true for all nontrivial two-component links? (Hint: Compute the Conway polynomial of the four links pictured below.)

3.6. The calculations of Exercise 3.5 show that by simultaneously taking the mirror image and reversing the orientation of one component of a nontrivial two-component link we change the sign of its Conway polynomial. Is this true for all nontrivial two-component links?

28

3. The Arithmetic of Knots

3.7. Compute the Conway polynomial of the connected sum of the right trefoil and the eight knot.

Lecture 4

Some Simple Knot Invariants

In this lecture, we define several knot invariants, which take values in the natural numbers and have very simple definitions, but unfortunately are hopelessly hard to compute, and so are not of any practical use in knot theory. One of them, the “genus” of a knot, has a deep geometric meaning (related to oriented surfaces spanning the knot in 3-space), but is also difficult to calculate even for the simplest knots. We also introduce the notion of tricolorability, which does have a practical purpose, as it allows to prove that many knots are nontrivial. Unfortunately, it takes values in ℤ/2ℤ = {𝑦𝑒𝑠, 𝑛𝑜} and does not help distinguishing nonisotopic nontrivial knots.

4.1. Stick number By definition, the stick number of a knot is the least number of rectilinear edges needed to construct the given knot. It is obviously an ambient isotopy invariant of knots. Figure 4.1 shows that the stick number of the trefoil is ≤ 6 and of the eight knot is ≤ 7. Actually, the stick number of the trefoil equals 6; this can be proved by a tedious case-by-case argument. For knots more complicated than the trefoil, finding the exact value of the stick number becomes hopelessly difficult. Thus the stick number may be a simple and nice invariant, but it is not practically useful for distinguishing nonequivalent knot diagrams. 29

30

4. Some Simple Knot Invariants 1 1 7 4 5 6

2

3

4

2 5 6 3

Figure 4.1. The stick number of the trefoil is ≤ 6 and of the eight knot is ≤ 7.

4.2. Crossing number The crossing number of a knot is defined as the least number of crossings in any knot diagram representing thegiven knot. It is obviously an ambient isotopy invariant of knots. Figure 4.1 shows that the crossing number of the trefoil is ≤ 3. Actually, the crossing number of the trefoil equals 3; this can be proved by showing that any knot with 2 crossings or less is trivial. However, for knots more complicated than the trefoil, finding the exact value of the crossing number becomes increasingly difficult. Figure 4.2 shows three knots with crossing numbers 5, 5, and 6. 51

52

61

Figure 4.2. The knots 51 , 52 , and 61

However, if the knot diagram is alternating and reduced, then the number of its crossings is automatically minimal — this is a recently

4.3. Unknotting number

31

proved classical conjecture that we accept without proof; a knot is called alternating if overpasses and underpasses alternate as we go around the knot, and reduced if it has no crossings of the type shown in gray in Figure 4.3.

Figure 4.3. A nonreduced knot

Because of the uniqueness of decomposition into primes, in order to classify knots, it suffices to classify prime knots. This has been done for knots with 16 crossings or less, first by hand (for small crossing numbers), and then by computer. For a small number (≤ 10) of crossings, it is customary to classify knots by listing them (as pictures) in a knot table in increasing order of their crossing number. This is done in Rolfsen’s classical (and beautiful) knot table (look it up in the Internet!). Here we present only a small table (7 crossings or less) in Figure 4.4. Traditionally, only one picture of a knot that does not coincide with its mirror image is shown in the table. Thus the table in Figure 4.4 shows the left trefoil, but not the right trefoil.

4.3. Unknotting number The unknotting number of a knot diagram is defined as the least number of crossing changes needed to transform the given knot into the trivial one. The fact that any knot can be trivialized by crossing changes is the assertion of the Trivialization Lemma 2.1. It is easy to see that the trefoil and the eight knot have unknotting numbers equal to 1. For knots with a large number of crossings, finding the unknotting number is a tedious task and many different knots have the same unknotting number, so that this invariant is not of any use for distinguishing nonisotopic knots, but it is rather curious and has been seriously studied by knot theorists, especially those interested in the unknotting problem.

32

4. Some Simple Knot Invariants

01

31

41

51

52

61

62

63

71

72

73

74

75

76

77

Figure 4.4. Knot table for knots with ≤ 7 crossings

4.4. Tricolorability A knot is called tricolorable if it possesses a knot diagram whose “strands” can be colored in three colors so that in the vicinity of every crossing point either all three colors are present or only one appears. Figure 4.5 shows that the trefoil is tricolorable, whereas the eight knot isn’t. + +

+

+ !

+

!

Figure 4.5. Coloring the trefoil and the eight knot

Theorem 4.1. Tricolorability is an invariant of knots, and any tricolorable knot requiring three colors is nontrivial.

4.5. Digression about orientable surfaces

33

Proof. The first assertion easily follows from the Reidemeister Theorem 1.1 — one easily checks that all three Reidemeister moves preserve tricolorability. The second assertion follows from the first one and from the obvious fact that the simplest diagram of the unknot (the round circle) is not tricolorable. □ The converse statement to Theorem 4.1 is not true, as Figure 4.3 shows (the eight knot is nontrivial, but not tricolorable).

4.5. Digression about orientable surfaces At this stage the reader should know (or learn by reading this section) something about orientable surfaces. By definition, a surface (a.k.a. a compact connected 2-dimensional manifold) is a compact connected topological space each point of which has a neighborhood homeomorphic to the open disk int(𝔻2 ) ≔ {(𝑥, 𝑦) ∈ ℝ2 ∣ 𝑥2 + 𝑦2 < 1}. A surface is orientable if it does not contain a Möbius band. An example of a nonorientable surface is the Klein bottle. Orientable surfaces can be homeomorphically embedded in ℝ3 . By definition, a surface-with-boundary is a compact topological space each point of which has a neighborhood homeomorphic either to the open disk or to the open half disk {(𝑥, 𝑦) ∈ ℝ2 ∣ 𝑥2 + 𝑦2 < 1, 𝑦 ≥ 0}, the set of the latter points (called boundary points and denoted by 𝜕𝑁); it is not hard to show that 𝜕𝑁 consists of a finite number of components, each of which is a topological circle) is non empty. Examples of surfaceswith-boundary are the cylinder 𝕊1 × [0, 1] and the disk 𝔻2 ≔ {(𝑥, 𝑦) ∈ ℝ2 ∣ 𝑥2 + 𝑦2 ≤ 1}. The cylinder has two boundary components, the disk, one. We will be interested in the case when the boundary consists of exactly one topological circle. A triangulation of a surface 𝑀 is its representation in the form 𝑁

𝑀=

⋃ 𝑖=1

𝜎𝑖2 ,

34

4. Some Simple Knot Invariants

where the 𝜎𝑖2 are 2-dimensional simplices (topological triangles) whose pairwise intersections are the empty set, or a common vertex, or a common side. A simple example of a triangulated sphere is the boundary of a tetrahedron. Other examples will be treated in the exercises. Fact 4.1. Any surface can be triangulated. Fact 4.2. The Euler characteristic 𝜒(𝑀) of a triangulated surface 𝑀, defined as 𝜒(𝑀) ∶ = 𝑉 + 𝐸 + 𝐹, where 𝑉 is the number of vertices, 𝐸 is the number of edges (sides), and 𝐹 is the number of faces (triangles), is a topological invariant of surfaces, i.e., homeomorphic surfaces have equal Euler characteristics. Fact 4.3. Any orientable surface is homeomorphic to exactly one of the following surfaces: the sphere 𝕊2 , the torus 𝕋2 , the sphere with two handles 𝕄22 , . . ., the sphere with 𝑘 handles 𝕄2𝑘 , . . .. These surfaces are classified by their Euler characteristic, which equals 2, 0, −2, . . ., 2 − 2𝑘, . . . respectively. They are shown in Figure 4.6.

,

,

, ...

...

, 11 11 11

22 22 22

, ... kk kk kk

Figure 4.6. List of all orientable surfaces

The genus 𝑔(𝑀) of an oriented surface 𝑀 is the number of its handles; since the sphere 𝕊2 has zero handles, the torus 𝕋2 , one handle, 𝕄2𝑛 has 𝑛 handles, so that these surfaces are of genus 0, 1, . . . , 𝑛, respectively. It follows from Fact 4.3 that the genus of an oriented surface 𝑀 isrelated to its Euler characteristic by the formula 𝜒(𝑀) = 2 − 2𝑔(𝑀).

4.6. Seifert surface of a knot

35

For a surface-with-boundary 𝑁 with a single boundary component, we have 𝜒(𝑁) = 1 − 2𝑔(𝑁).

4.6. Seifert surface of a knot The Seifert surface of a knot is defined as an orientable surface-withboundary in ℝ3 whose boundary is the given knot. Figure 4.7 shows the knot 52 (a) and its Seifert surface (c), while (b) shows how the surface can be constructed. The reader can actually model this construction by using paper (colored in different colours on opposite sides), scissors, and Scotch tape.

(a)

(b)

(c)

Figure 4.7. Seifert surface for the 52 knot

Figure 4.8 shows the construction of the Seifert surface of the trefoil, a paper model of which can also be easily made.

31 Figure 4.8. Seifert surface for the trefoil

36

4. Some Simple Knot Invariants

4.7. The genus of a knot By definition, the genus of a knot is the genus of its Seifert surface with the minimal number of handles. The genus of a knot is obviously a knot invariant. Once a Seifert surface 𝑆 with the minimal number of handles of the given knot 𝐾 has been constructed, the genus 𝑔(𝐾) of the knot, which equals (by definition) the genus 𝑔(𝑆) of the surface, can easily be calculated if one knows its Euler characteristic 𝜒(𝑆) is, because the following formula holds: (1 − 𝜒(𝑆)) . 2 But we will not go into details here, because although the genus of a knot may be an interesting invariant, it not particularly useful for distinguishing (= classifying) knots, just as all the other invariants mentioned in this lecture. 𝑔(𝐾) = 𝑔(𝑆) =

4.8. Exercises 4.1. Give a lower bound for the stick number of the knot 51 . What is your conjecture for the value of the stick number of that knot? 4.2. What is your conjecture of the value of the stick number of the eight knot 41 ? Explain how would you go about proving it (without going into details). 4.3. Prove that any knot diagram with two crossing points or less is trivial. 4.4. Prove that the only two knots with crossing number 3 are the two trefoils. 4.5. Find the unknotting number of the knots 51 , 52 , and 61 . 4.6. Which of the knots 51 , 52 , and 61 are tricolorable? 4.7. On the classical model of the torus (square with identified opposite sides), draw a triangulation of the torus. 4.8. Indicate how the sides of a regular octagon can be identified so as to obtain a sphere with two handles. 4.9. Indicate how the sides of a regular 4𝑘-gon can be identified so as to obtain a sphere with 𝑘 handles.

4.8. Exercises 4.10. Draw a nice picture of the Seifert surface of the eight knot. 4.11. What is the genus of the eight knot? 4.12. What is the genus of the 52 knot?

37

Lecture 5

The Kauffman Bracket

In this lecture, we study the Kauffman bracket. In our course, it plays an important, but auxiliary role: it is needed only to define the famous Jones polynomial (this will be done in the next lecture). The Kauffman bracket, like the Conway polynomial, assigns a polynomial to each link diagram, but the links in question are assumed nonoriented. The Kauffman bracket is a fundamental tool in physics, more important than the Jones polynomial (in particular in quantum field theory), although the Jones polynomial also has a significant relationship with physics (2-dimensional statistical models).

5.1. Digression: statistical models in physics In this section, we give a rough idea of the notion of 2-dimensional statistical model, which has no direct bearing on knot theory, but will serve, nevertheless, as the inspiration for the definition of the main protagonist of this lecture on the Kauffman bracket, which will be a kind of phony “partition function” of a 2-dimensional “state model” determined by the given knot diagram. Let me explain what the words in quotation marks mean. Roughly speaking, a 2-dimensional statistical (or state) model is a system consisting of huge number of particles {𝑝 𝑖 } = 𝑃, represented by points in the plane and joined by lines which indicate interactions between particles. The particles can be in one of two states (spins): “spin 39

40

5. The Kauffman Bracket

up” and “spin down” (traditionally shown by vertical vectors pointing up and down, respectively) and the state of the system is a picture of the particles supplied with spins and their interactions — see Figure 5.1.

Figure 5.1. A state of a small part of a 2-dimensional statistical model

A given model 𝑃 consisting of 𝑛 particles obviously has 2𝑛 states, denoted 𝑠 ∈ 𝒮(𝑃). The main characteristic of such a model, called its partition function, is given by a formula such as 𝑍(𝑃) = ∑ exp( 𝑠∈𝒮(𝑃)

−1 ∑ 𝜀(𝑠(𝑝 𝑖 ), 𝑠(𝑝𝑗 ))), 𝑘𝑇 𝑝 ,𝑝 ∈𝑃 𝑖

𝑗

where 𝜀( ⋅ , ⋅ ) is a real-valued function expressing the “energy” of interaction between two interacting (i.e., joined by a line in the picture) particles, 𝑇 is “temperature”, and 𝑘 is the “Boltzmann constant”. We shall not explain the meaning of the words in quotation marks in the above sentence — all we need to know is that the formula is the sum over all states of the product of something called interactions (the product — because the exponential of a sum equals the product of exponentials of the summands).

5.2. The “state” of a (nonoriented) knot diagram

41

5.2. The “state” of a (nonoriented) knot diagram Suppose we consider a crossing point of a nonoriented knot (or link). Unlike crossing points of an oriented knot, among crossing points of nonoriented ones, we cannot distinguish positive and negative crossings, as we did in the Conway skein relation. However, at each crossing point, there are two vertical angles of two different kinds, that we call A-angles and B-angles (see Figure 5.2(a)).

Figure 5.2. A-angles and B-angles and a state of a knot

A-angles are characterized by the fact that an observer moving along the overpass in any direction first sees the A-angle to their right, and then (after passing the crossing point), to their left. For B-angles, it’s the other way around: an observer moving along the overpass first sees the B-angle to the left, and then (after passing the crossing point), to the right. Now let us consider a (nonoriented) link diagram (such as the one in Figure 5.2). At each of the crossing points, let us choose either an A-angle or a B-angle, and indicate the chosen angle by drawing a short “stick” in it. Such a choice is called a state of the link diagram (look at Figure 5.2(b)). If the link diagram has 𝑛 crossing points, there will be 2𝑛 possible states. In the figure,we have shaded the A-angles, and our picture acquired a chessboard coloring.

42

5. The Kauffman Bracket

Now let us look at a possible state of a simpler, more familiar knot, say the eight knot (Figure 5.3(a)).

Figure 5.3. A state of the eight knot

For a given state 𝑠 ∈ 𝒮, let us denote the number of chosen A-angles by 𝛼(𝑠), and the number of B-angles by 𝛽(𝑠). Obviously, 𝛼(𝑠) = 𝑛 − 𝛽(𝑠). Now let us “smooth out” the crossing points as shown in Figure 5.3(b). We then obtain a certain number of topological circles; this number is denoted by 𝛾(𝑠). In our example, 𝛼(𝑠) = 3, 𝛽(𝑠) = 1, 𝛾(𝑠) = 2.

5.3. Definition and properties of the Kauffman bracket The Kauffman bracket assigns to a each link diagram 𝐾 a polynomial with integer coefficients in three variables 𝑎, 𝑏, 𝑐, denoted by ⟨𝐾⟩ and defined by the formula ⟨𝐾⟩ ∶ = ∑ 𝑎𝛼(𝑠) 𝑏𝛽(𝑠) 𝑐𝛾(𝑠)−1 . 𝑠∈𝒮

Like the formula for the partition function, here we have a sum of certain products over all states. The calculation of the value of ⟨ ⋅ ⟩ in our example (the eight knot) is the object of Exercise 5.3. It easily follows from the definition that the Kauffman bracket possesses the following properties. (I) Normalization: ⟨○⟩ = 1.

5.4. Is the Kauffman bracket invariant?

43

(II) Skein relation: ⟨

⟩ = 𝑎⟨

⟩ + 𝑏⟨

⟩.

=

=

The skein relation for the Kauffman bracket differs from the Conway skein relation. It should be understood as follows: we have three nonoriented link diagrams 𝐿× , 𝐿= , 𝐿 that are identical outside three small disks, and are as shown in the figure inside the disks; the relation says that the bracket of the diagram with a crossing point inside the disk is equal to the sum of the two other brackets with coefficients 𝑎 and 𝑏, i.e., we have ⟨𝐿× ⟩ = 𝑎⟨𝐿= ⟩ + 𝑏⟨𝐿 ⟩. (III) Adding the unknot: ⟨𝐿 ⊔ ○⟩ = 𝑐 ⋅ ⟨𝐿⟩. Here the left-hand side of the equality 𝐿 ⊔ ○ is the link diagram consisting of the link 𝐿 and a topological circle that does not have any crossing points with 𝐿. The equality shows that adding such a circle to a link results in its Kauffman bracket being multiplied by 𝑐.

5.4. Is the Kauffman bracket invariant? Let us see if the Kauffman bracket is invariant w.r.t. the Reidemeister moves Ω2 , Ω3 , Ω1 . If it is invariant w.r.t. all three, it will follow by the Reidemeister Theorem that ⟨ ⋅ ⟩ is an isotopy invariant. We begin with Ω2 , and consider an arbitrary link which has, inside a little disk, two arcs, one of which overpasses the other twice. Our goal is to check that its Kauffman bracket is equal to the bracket of the same link, but with two nonintersecting arcs in the little disk. First, we apply the skein relation (II) to the upper crossing point inside the disk under consideration, then apply it to the lower crossing

44

5. The Kauffman Bracket

points of the two new disks, obtaining

⟨𝐾⟩ = ⟨

⟩ = 𝑎⟨

⟩ + 𝑏⟨

⟩ + 𝑎2 ⟨

= 𝑎𝑏⟨

+ 𝑏2 ⟨

⟩

⟩

⟩ + 𝑎𝑏⟨

⟩.

We now apply property (III) to the second summand of the last line, and after gathering like terms, we have

⟨𝐾⟩ = ⟨

⟩

= (𝑐 + 𝑎2 + 𝑏2 )⟨

⟩ + 𝑎𝑏⟨

⟩.

The result is not what we wanted, but if we simplify the bracket by putting 𝑏 = 𝑎−1 and 𝑐 = −𝑎2 − 𝑎−2 , we obtain the desired result — namely the (simplified) bracket of our link — with the distinguished disk containing two nonintersecting arcs. From now on, by abuse of notation, we shall denote the simplified bracket (which is now a Laurent polynomial in the variable 𝑎) by the same symbol as the old one.

5.4. Is the Kauffman bracket invariant?

45

The proof of the Ω3 -invariance of the (simplified) bracket appears in the figure below.

⟨

⟩ = 𝑎⟨

⟩ + 𝑎−1 ⟨

⟩

⟨

⟩ = 𝑎⟨

⟩ + 𝑎−1 ⟨

⟩=

(1)

(3)

(2)

The first line of the equation should be read from left to right (we apply the skein relation again), the second line, from right to left, after having noted that the equality between the middle summands is obtained by applying Ω2 twice (which is legal, as we have just shown). It now suffices to prove that our bracket is Ω1 -invariant. We have ⟩ + 𝑎−1 ⟨

⟩ = 𝑎⟨

⟨

= −𝑎3 ⟨

⟩

⟩.

Thus we see that the Kauffman bracket is not Ω1 -invariant. For the other little loop, we have similarly ⟨

⟩ = −𝑎−3 ⟨

⟩.

Let us summarize our results in the following theorem. Theorem 5.1. The (simplified) Kauffman bracket ⟨ ⋅ ⟩ has the following properties: =

(I) ⟨○⟩ = 1; (II) ⟨𝐿× ⟩ = 𝑎⟨𝐿= ⟩ + 𝑎−1 ⟨𝐿 ⟩;

46

5. The Kauffman Bracket (III) ⟨𝐿 ⊔ ○⟩ = −(𝑎2 + 𝑎−2 ) ⋅ ⟨𝐿⟩; (IV) ⟨ ⋅ ⟩ is Ω2 - and Ω3 -invariant;

(V) ⟨ ⋅ ⟩ is not Ω1 -invariant, the application of the Ω1 -move results in the multiplication of the bracket by the coefficient (−𝑎)±3 , where the sign depends of the type of the disappearing little loop.

5.5. Exercises 5.1. Compute the Kauffman bracket of the following knot diagrams.

5.2. Compute the Kauffman bracket of the knot diagram of the right trefoil shown in the knot table (Figure 4.4). 5.3. Compute the Kauffman bracket of diagram of the eight knot shown in Figure 5.3. 5.4. Compute the Kauffman bracket of the knot diagram of the knot 52 shown in the knot table (Figure 4.4). 5.5. Compute the Kauffman bracket of the knot diagram of the knot 51 shown in the knot table (Figure 4.4). 5.6. Give a detailed proof of property (II) of the Kauffman bracket (the Kauffman skein relation). 5.7. Give a detailed proof of property (III) of the Kauffman bracket. 5.8. Show that the Kauffman bracket is mutiplicative w.r.t. the connected sum of knots. 5.9. Compute the Kauffman bracket of the granny knot. (Hint: use the results of Exercises 5.2 and 5.8.)

Lecture 6

The Jones Polynomial

The Jones polynomial was invented by Vaughan Jones in 1985. It is a powerful knot and link invariant. Unlike the Conway polynomial, it distinguishes a knot 𝐾 from its mirror image 𝐾 ∗ whenever 𝐾 is not isotopic to 𝐾 ∗ . However, it is not a complete invariant. Jones’s original definition was based on deep topological and algebraic constructions and facts, namely the Markov theorem on the closure of braids and the Ocneanu trace in the Temperley–Lieb algebra. Our exposition, however, is elementary — it follows the work of Louis Kauffman and is based on the Kauffman bracket.

6.1. Definition via the Kauffman bracket Let 𝐿 be an oriented link or knot. Any link diagram has a finite number of crossing points. In the definition of the Conway polynomial, we distinguish positive and negative crossings (shown in Figure 6.1(a) and (b), respectively). Using this distinction here, let us number the the crossings points of 𝐿 and set 𝜀(𝑃𝑖 ) = +1 if the 𝑖th crossing point 𝑃𝑖 is positive and 𝜀(𝑃𝑖 ) = −1 if it is negative, and define the writhe 𝑤(𝐿) of an oriented link diagram 𝐿 by setting 𝑤(𝐿) ∶ =

∑

𝜀(𝑃𝑖 ).

{crossings points 𝑃𝑖 }

47

48

6. The Jones Polynomial

(a)

(b)

Figure 6.1. Positive and negative crossings

Lemma 6.1. The writhe 𝑤( ⋅ ) is Ω2 - and Ω3 -invariant. Under the Ω1 move, it changes by −1 (resp. +1) when the disappearing crossing point is positive (resp. negative). The proof is the object of Exercise 6.1. Given an oriented link diagram 𝐿, we denote by |𝐿| the same link, but without the orientation. Recall that in the previous lecture, we defined and learned to calculate the (simplified) Kauffman bracket ⟨|𝐿|⟩. We can now define the (preliminary version) 𝐽( ⋅ ) of the Jones polynomial by setting 𝐽(𝐿) ∶ = (−𝑎)−3𝑤(𝐿) ⟨|𝐿|⟩. Thus the Jones polynomial assigns to any oriented link 𝐿 a Laurent polynomial 𝐽(𝐿) ∈ ℤ[𝑎, 𝑎−1 ]. Theorem 6.1. The Jones polynomial 𝐽( ⋅ ) (in its preliminary version) is an ambient isotopy invariant. Sketch of the proof. It suffices to check that 𝐽( ⋅ ) is invariant w.r.t. the three Reidemeister moves. The fact that it is Ω2 - and Ω3 -invariant immediately follows from Lemma 6.1 and item (IV) of Theorem 5.1. Its Ω1 -invariance is less obvious, and is the object of Exercise 6.3. □ Remark 6.1. To pass from the preliminary version 𝐽 of the Jones polynomial to its final (= usual) version 𝑉, it suffices to make the change of variables 𝑎 = 𝑞−1/4 . We will do this a little later, after we have performed some calculations with the preliminary version in order to establish the main properties of the Jones polynomial. (These calculations are more conveniently performed using the variable 𝑎 than the variable 𝑞.)

6.2. Main properties of 𝐽( ⋅ )

49

6.2. Main properties of 𝐽( ⋅ ) (I𝐽 ) Normalization: 𝐽(○) = 1. This immediately follows from item (I) of Theorem 5.1. (II𝐽 ) Skein relation for the Jones polynomial: 𝑎4 𝐽(𝐿+ ) − 𝑎−4 𝐽(𝐿− ) = (𝑎−2 − 𝑎2 )𝐽(𝐿0 ), or in the standard symbolic form 𝑎4 𝐽(

) − 𝑎−4 𝐽(

) = (𝑎−2 − 𝑎2 )𝐽(

)

where 𝐿+ , 𝐿− , 𝐿0 are three oriented links, identical outside three little disks, and are as shown inside the disks. To prove property (II𝐽 ), we begin by writing out the skein relation for the Kauffman bracket twice ⟨

⟩ = 𝑎⟨

⟨

⟩ = 𝑎−1 ⟨

⟩ + 𝑎−1 ⟨

⟩,

⟩ + 𝑎⟨

⟩.

then multiply the first equality by −𝑎−1 , the second by 𝑎, and add the resulting equations, obtaining ⟩ − 𝑎−1 ⟨

𝑎⟨

⟩ = (𝑎2 − 𝑎−2 )⟨

⟩.

Now let us denote by 𝐿+ , 𝐿− , 𝐿∘ the oriented links obtained from the nonoriented links in the above relation by orienting them as shown in Figure 6.2.

L+

L−

Figure 6.2. The oriented links 𝐿+ , 𝐿− , 𝐿∘

We then have 𝑎⟨|𝐿+ |⟩ − 𝑎−1 ⟨|𝐿− |⟩ = (𝑎2 − 𝑎−2 )⟨|𝐿∘ |⟩,

L◦

50

6. The Jones Polynomial

where | ⋅ |, as above, indicates that we are considering a nonoriented link. By definition of the writhe, we have 𝑤(𝐿± ) = 𝑤(𝐿∘ ) ± 1, and recalling the definition of the polynomial 𝐽( ⋅ ), we obtain 𝑎(−𝑎3 )𝐽(𝐿+ ) − 𝑎−1 (−𝑎)−3 𝐽(𝐿− ) = (𝑎2 − 𝑎−2 )𝐽(𝐿∘ ), which is exactly the required relation. (III𝐽 ) Adding an unknot: 𝐽(𝐿 ⊔ ○) = −(𝑎2 − 𝑎−2 )𝐽(𝐿). The proof of this statement is the object of Exercise 6.4.

6.3. Axioms for the Jones polynomial We now make the substitution 𝑎 = 𝑞−1/4 , obtaining the Jones polynomial in its usual form 𝑉(𝐿) ∶ = 𝐽(𝐿)|𝑎=𝑞−1/4 . The main properties of the Jones polynomial 𝑉( ⋅ ) immediately follow from the corresponding properties of the polynomial 𝐽( ⋅ ) proved above. As we shall soon see, these properties may be regarded as axioms for the Jones polynomial. (0) Invariance: The Jones polynomial 𝑉(𝐿) ∈ ℤ[𝑞1/2 , 𝑞−1/2 ] of any oriented link (in particular, of any oriented knot) 𝐿 is an (ambient) isotopy invariant. (I) Normalization: 𝑉(○) = 1. (II) Skein relation for the Jones polynomial: 𝑞𝑉(

) − 𝑞−1 𝑉(

) = (√𝑞 −

1 )𝑉( √𝑞

).

(III) Adding an unknot: 𝑉(𝐿 ⊔ ○) = −(

1 + √𝑞)𝑉(𝐿). √𝑞

Theorem 6.2. The Jones polynomial 𝐽( ⋅ ) is an (ambient) isotopy invariant of oriented links satisfying axioms (I), (II), (III) and is uniquely determined by these axioms.

6.4. Multiplicativity

51

Proof. The existence of the polynomial 𝐽(𝐿) satisfying axioms (I)–(III) was proved above. Let us prove uniqueness. We shall prove this by induction on the number 𝑘 of crossings. If 𝑘 = 0, then 𝐿 is a trivial link, say with 𝑚 components. In that case, its Jones polynomial can be computed by successively applying property (III𝐽 ). Its actual value is not important for the proof — it is the object of Exercise 6.5. Assuming uniqueness for 𝑘 < 𝑛, let us prove it for 𝑘 = 𝑛. We shall prove this for a fixed 𝑛 by induction on the number 𝑙 of crossing changes needed to trivialize the given link diagram 𝐿 (such a finite number exists by the Trivialization Lemma 2.1). If 𝑙 = 0, then 𝐿 is a trivial link, and we know what 𝐽(𝐿) is equal to. So we assume that uniqueness has been proved for links that can be trivialized by 𝑙 crossing changes. Let us prove it for 𝑙 + 1. Applying axiom (II) to any crossing point of the given link 𝐿, we can write 𝑞𝑉(

) − 𝑞−1 𝑉(

) = (√𝑞 −

1 )𝑉( √𝑞

).

Of the two summands on the left-hand side, at least one can be trivialized by 𝑙 crossing changes, and so its value is well defined by the induction hypothesis on 𝑙. The link diagram appearing in the right-hand side has 𝑛 − 1 crossing points, and so the value of the right-hand side is known inductively. Therefore, we can calculate the unknown term in the lefthand side and thus obtain the value of the Jones polynomial of any link diagram with 𝑛 crossings that can be trivialized by 𝑙+1 crossing changes. Thus induction on 𝑘 and 𝑙 concludes the proof of Theorem 6.2. □

6.4. Multiplicativity The Jones polynomial behaves very nicely w.r.t. the connected sum operation for knots, and there is nice formula for the Jones polynomial for the disjoint union of two links (by the disjoint union of two links, one means the link obtained by placing the given two links in different halfspaces and taking their union). Indeed, we have the following theorem.

52

6. The Jones Polynomial

Theorem 6.3. (a) With respect to the connected sum operation for knots, the Jones polynomial is multiplicative, i.e., 𝑉(𝐾1 # 𝐾2 ) = 𝑉(𝐾1 ) ⋅ 𝑉(𝐾2 ). (b) With respect to the disjoint union of links, the Jones polynomial behaves as follows: 1 𝑉(𝐿1 ⊔ 𝐿2 ) = (√𝑞 − )𝑉(𝐿1 ) ⋅ 𝑉(𝐿2 ). √𝑞 The proof of this theorem is the object of Exercise 6.7. Remark 6.2. There is no well-defined connected sum operation for multicomponent links, because there is no preferred way to choose the components that we are to connect. Nevertheless, we will use the notation 𝐿1 # 𝐿2 to indicate any one of the possible ways to connect the two links 𝐿1 and 𝐿2 (by specifying one component in each link). In that notation, we have 𝑉(𝐿1 # 𝐿2 ) = 𝑉(𝐿1 ) ⋅ 𝑉(𝐿2 ). The proof of this formula is the object of Exercise 6.6.

6.5. Chirality and reversibility Theorem 6.4. (a) Reversing the orientation of all the components of a link diagram does not change its Jonespolynomial. (b) The Jones polynomial of the mirror image of a link diagram 𝐿 is obtained from the Jones polynomial of 𝐿 by substituting 𝑞−1 for 𝑞. The proofs of the two assertions of the theorem are the object of Exercises 6.14 and 6.13, respectively.

6.6. Is the Jones polynomial a complete invariant? The Jones polynomial, unlike the Conway polynomial, distinguishes the left trefoil from the right trefoil, as the reader can verify by doing Exercise 6.2. Is it a complete invariant, i.e., do we have the implication 𝑉(𝐿) = 𝑉(𝐿′ ) ⟹ 𝐿 ∼ 𝐿′ ? The answer is “no” — a simple counterexample, obtained by taking connected sums of links in different ways, appears in Exercise 6.9.

6.7. Is 𝑉 a Laurent polynomial in 𝑞?

53

Vaughan Jones conjectured that his polynomial is a complete invariant for prime knots, but his conjecture was very quickly refuted by several knot theorists. A counterexample appears in Figure 6.3.

Figure 6.3. Nonequivalent prime knots with same Jones polynomial

The proof of the fact that the two knots in the above figure have the same Jones polynomial is the object of Exercise 6.8. The fact that the two knots are not equivalent uses the so-called Arf invariant and is not given in this course.

6.7. Is 𝑉 a Laurent polynomial in 𝑞? The answer is given by the following theorem. Theorem 6.5. (a) If the number of components of an oriented link 𝐿 is odd (in particular, if it is a knot), then 𝑉(𝐿) ∈ ℤ[𝑞, 𝑞−1 ], i.e., 𝑉(𝐿) contains only terms of the form 𝑞𝑘 , 𝑘 ∈ ℤ. (b) If the number of components of an oriented link 𝐿 is even, then 𝑉(𝐿) contains only terms of the form 𝑞(2𝑘+1)/2 , 𝑘 ∈ ℤ. Proof. We know from Exercise 6.5 that the Jones polynomial of the trivial 𝑚-component oriented link is given by (−𝑞−1/2 − 𝑞1/2 )

𝑚−1

= 𝑞(𝑚−1)/2 (−𝑞−1 − 1)

𝑚−1

,

so that both statements of the theorem hold for trivial 𝑚-component links. We also know (from the proof of Theorem 6.2) that 𝑉(𝐿) can be

54

6. The Jones Polynomial

computed from 𝑉 of the trivial link by successively applying the Jones skein relation (II). Let us denote by 𝑚+ , 𝑚− , 𝑚0 the number of components of 𝐿+ , 𝐿− , 𝐿0 , respectively. To prove the theorem, it suffices to show that (1) the numbers 𝑚+ and 𝑚− have the same parity; (2) the numbers 𝑚+ and 𝑚0 have opposite parities. Statement (1) is obvious (actually 𝑚+ = 𝑚− ). To prove (2), it suffices to show that 𝑚+ = 𝑚0 ± 1. The proof of that equality is the subject of Exercise 6.10. □

6.8. Knot tables revisited As we mentioned in Lecture 3, because of the unique decomposition theorem into primes, in order to classify knots, it suffices to classify prime knots. For knots with a small number of crossings, say ≤ 9, prime knots (represented by their knot diagrams) are listed in knot tables. In these tables, only one picture of a knot that does not coincide with its mirror image is usually shown. The Jones polynomial has been calculated for all these knots, and it turned out that Jones polynomials of any pair of different prime knots with ≤ 9 crossings are different, which means that the Jones polynomial classifies prime knots with ≤ 9 crossings, so that Jones’s conjecture is true for these prime knots. For larger values of the number of crossings, this is no longer true, as we saw above (Figure 6.3). The number of knots with given crossing number 𝑐 grows exponentially with 𝑐, in particular, there are 3 prime knots with 6 crossings (as can be seen in the knot table in Figure 4.4), 166 with 10 crossings, and 1 388 705 with 16 crossings. For this reason, for a large number of crossings, the computer is used to produce knot tables. In them, knots are not presented as pictures, but by a special numerical encoding known as the Dowker–Thistlethwaite code of the knot. To generate the Dowker–Thistlethwaite code, move along the knot using an arbitrary starting point and direction. Label each of the 𝑛 crossings with the numbers 1, . . . , 2𝑛 in order of traversal (each crossing is visited and labelled twice), with the following modification: if the label is an even number and the strand followed is an overcrossing, then

6.9. Exercises

55

change the sign on the label to negative. When finished, each crossing will be labelled a pair of integers, one even and one odd. The Dowker– Thistlethwaite notation notation is the sequence of even integer labels associated with the labels 1, 3, . . . , 2𝑛 − 1 in turn. For example, a knot diagram may have crossings labelled with the pairs (1, 6), (3, −12), (5, 2), (7, 8), (9, −4), (11, −10). Then the Dowker– Thistlethwaite code for this labelling will be the following sequence 6, −12, 2, 8, −4, −10 (or any of its cyclic permutations). It can be shown that any prime knot is uniquely determined by its Dowker–Thistlethwaite code. The current software that produces knot tables is called “Knotscape” and is due to Thistlethwaite, Weeks, and Hoste. There are also tables of prime links (their definition is the object of Exercise 6.11) for links with 13 crossings compiled by Thistlethwaite. The reader can try to construct prime link tables for links with a small number of crossings (Exercise 6.12).

6.9. Exercises 6.1. Prove Lemma 6.1 in the general case of multicomponent links. 6.2. Compute the (preliminary versions of the) Jones polynomials of the two trefoils. Are they isotopic? 6.3. Prove the Ω1 -invariance of the Jones polynomial 𝐽( ⋅ ) (preliminary version). 6.4. Prove property (III𝐽 ) of the Jones polynomial 𝐽. 6.5. Compute the Jones polynomial of the trivial 𝑚-component link and 𝑚−1 show that it equals (−𝑞−1/2 − 𝑞1/2 ) . 6.6. Prove the formula in Remark 6.2. 6.7. Prove Theorem 6.3. (Hint: Use the definition of the Jones polynomial via the Kauffman bracket and the state sum definition of the Kauffman bracket.) 6.8. Show that the two knots in Figure 6.3 have the same Jones polynomial. (Hint: You need not explicitly compute the Jones polynomials of the two knots — it suffices to order the crossing points to which you apply the skein relation so that it yields the same result at each step.)

56

6. The Jones Polynomial

6.9. Show that the two links in the figure below have identical Jones polynomials, but are not ambient isotopic.

6.10. Show that 𝑚+ = 𝑚0 ± 1 in the proof of Theorem 6.5. 6.11. Define the notion of prime link and give examples (if any) of prime 3-component links. 6.12. Compile a table of prime links with ≤ 5 crossings. 6.13. Prove that the Jones polynomial of the mirror image 𝐿∗ of a link diagram 𝐿 is obtained from the Jones polynomial of 𝐿 by replacing 𝑞 by 𝑞−1 . 6.14. Prove that the Jones polynomial of an oriented link is unchanged if the orientations of all its components are reversed. 6.15. Compute the Jones polynomial of the eight knot in two ways: by using the definition (via the Kauffman bracket) and by using the axioms of the Jones polynomial (including the Jones skein relation). 6.16. Draw a picture of the knot with Dowker–Thistlethwaite code 6, −12, 2, 8, −4, −10. 6.17. Write down the Dowker–Thistlethwaite code of the eight knot.

Lecture 7

Braids

Braids, like knots and links, are curves in ℝ3 with a natural composition operation. They have the advantage of being a group under the composition operation (unlike knots, all braids have inverses). The braid group is denoted by 𝐵𝑛 . The subscript 𝑛 is a natural number, and there is an infinite series of nested braid groups

𝐵1 ⊂ 𝐵2 ⊂ . . . ⊂ 𝐵𝑛 ⊂ . . .

At first, we shall define and study these groups geometrically, then look at them as purely algebraic objects via their presentation obtained by Emil Artin (who actually invented braids in the 1920-ies). Then we shall see that braids are related to knots and links via a geometric construction called “closure” and study the consequences of this relationship. We note at once that braids play an important role in many fields of mathematics and theoretical physics, exemplified by such important terms as “braid cohomology”, “braided vector space”, “braided monoidal category”, “braided Hopf algebra”, and this is not due to their relationship to knots. But here we are interested in braids mainly because of this relationship. 57

58

7. Braids

7.1. Geometric braids A braid in 𝑛 strands is a set consisting of 𝑛 pairwise nonintersecting polygonal curves (called strands) in ℝ3 joining 𝑛 points aligned on a horizontal line 𝐿 to 𝑛 points with the same 𝑥, 𝑦 coordinates aligned on a horizontal line 𝐿′ parallel to 𝐿 and located lower than 𝐿; the strands satisfy the following condition: when we move downward along any strand from a point on 𝐿 to a point on 𝐿′ , the tangent vectors of this motion cannot point upward (i.e., the 𝑧-coordinate of these vectors is always nonpositive).

Figure 7.1. Examples of braids

Two braids are called equivalent (or isotopic) if there is a sequence of braids in which every braid is obtained from the previous one by a Δ-move. Two braids 𝑏 and 𝑏′ with the same number of strands have a natural composition, consisting in identifying the 𝑛 lower endpoints of 𝑏 with the upper endpoints of 𝑏′ (see Figure 7.2 for the case 𝑛 = 4). It is easy to see that the composition operation is well defined on isotopy classes of braids. In what follows, we will use the term braid on 𝑛 stands (or simply 𝑛-braid or braid) both for concrete geometric braids and for equivalence classes — the reader will understand what is meant from the context.

7.2. The geometric braid group 𝐵𝑛

×

59

=

Figure 7.2. Composition of braids

7.2. The geometric braid group 𝐵𝑛 Theorem 7.1. For any 𝑛 ≥ 1, the set (of equivalence classes) of braids forms a group, denoted by 𝐵𝑛 . Proof. The neutral element is the braid all of whose strands are straight vertical lines. The composition operation is obviously associative. Any braid has an inverse, namely its mirror image w.r.t. the horizontal plane containing its lower endpoints (this is clear from Figure 7.3).

Figure 7.3. Inverse braid

This completes the proof of the theorem.

□

It is easy to see that 𝐵1 = 0 and 𝐵2 ≅ ℤ. Is the group 𝐵𝑛 Abelian? The answer is “no”, provided 𝑛 ≥ 3; finding an example is easy (Exercise 7.1).

60

7. Braids

Is the group 𝐵𝑛 finitely generated? The answer is yes, 𝐵𝑛 has 𝑛 − 1 canonical generators 𝑏1 , 𝑏2 , . . . , 𝑏𝑛−1 . They are shown on the left-hand side of Figure 7.4. 1

2

3

4

5 b2

b1

b−1 4 b3

b2

b1 b−1 3 bn−1

b4 =

−1 b2 b−1 4 b3 b1 b3 b4

(b)

(a) Figure 7.4. Braid generators

The right-hand side of the figure shows how a 5-braid can be expressed as a product of canonical generators. This construction is general.

7.3. Digression on group presentations Readers familiar with the notion of group presentation can skip this section and go on to the next one. Roughly speaking, a group presentation of a group 𝐺 is a method for defining the group by listing its generators and the relations that these generators must satisfy. Thus the free group in 𝑛 generators 𝐹𝑛 is presented in the form 𝐹𝑛 ↔ ⟨𝑔1 , 𝑔2 , . . . , 𝑔𝑛 ∣ ⟩, (following tradition, we do not explicitly indicate the trivial relations −1 𝑔𝑖 𝑔−1 𝑖 = 𝑔𝑖 𝑔𝑖 = 𝑒,

𝑔𝑗 𝑒 = 𝑒𝑔𝑗 = 𝑔𝑗 ,

which are satisfied in any group, so only the generators are indicated in the presentation of free groups). Thus the free group 𝐹𝑛 in 𝑛 generators is defined as the set of equivalence classes of words in the alphabet

7.3. Digression on group presentations

61

−1 −1 𝑔1 , 𝑔−1 1 , 𝑔2 , 𝑔2 . . . 𝑔𝑛 , 𝑔𝑛 , where two words are considered equivalent if one can be transformed into the other by means of the trivial relations. −1 For example, in 𝐹3 we have 𝑔1 = 𝑔2 𝑔−1 2 𝑔1 𝑔3 𝑔3 because −1 −1 −1 𝑔1 = 𝑒𝑔1 = (𝑔2 𝑔−1 2 )𝑔1 = (𝑔2 𝑔2 𝑔1 )𝑒 = (𝑔2 𝑔2 𝑔1 )(𝑔3 𝑔3 ).

The group of residues modulo 𝑛 is presented in the form ℤ/𝑛ℤ ↔ ⟨𝑔 ∣ 𝑔𝑛 = 𝑒⟩, while the direct sum of two copies of the integers is presented as ℤ ⊕ ℤ ↔ ⟨𝑔, ℎ ∣ 𝑔ℎ𝑔−1 ℎ−1 = 𝑒⟩. The formal definition of the group given by its presentation is as follows: a group presentation in the alphabet −1 −1 𝒜 = {𝑔1 , 𝑔−1 1 , 𝑔2 , 𝑔2 , . . . , 𝑔 𝑛 , 𝑔𝑛 }

of a group 𝐺 is an expression of the form 𝐺 ↔ ⟨𝑔1 , 𝑔2 , . . . , 𝑔𝑛 ∣ 𝑅1 = 𝑅2 = . . . = 𝑅𝑘 = 𝑒⟩, where the 𝑅𝑗 are words in the alphabet 𝒜, if 𝐺 ≅ 𝐹𝑛 /⟨⟨𝑅1 , . . . 𝑅𝑘 ⟩⟩, where ⟨⟨𝑅1 , . . . 𝑅𝑘 ⟩⟩ denotes the minimal normal subgroup containing the elements 𝑅1 , . . . 𝑅𝑘 . Although the presentation of groups provides us with a convenient way of performing calculations with elements of the group (for examples, see below), it doesn’t always help to identify the presented group (we will discuss this in Section 7.5 below). The presentation of a group 𝐺 ↔ ⟨𝑔𝑖 , 𝑅𝑗 ⟩ allows to perform calculations within the group by replacing 𝑅𝑖 by the neutral element 𝑒 (and vice versa) and using the trivial relations. Here is an example of a calculation in the group presented as ⟨𝑔, ℎ ∣ 𝑔ℎ𝑔−1 ℎ−1 ⟩: 𝑔ℎ𝑔−1 ℎ−1 = 𝑒 ⟹ (𝑔ℎ𝑔−1 ℎ−1 )(ℎ𝑔) = 𝑒(ℎ𝑔) ⟹ 𝑔ℎ𝑔−1 (ℎ−1 ℎ)𝑔 = 𝑒(𝑔ℎ) ⟹ 𝑔ℎ𝑔−1 𝑔 = ℎ𝑔, which means that the two elements ℎ𝑔 and 𝑔ℎ in ℤ ⊕ ℤ are equal, i.e., represent the same element of the group ℤ⊕ℤ. The reader should check these calculations carefully and identify the specific relations used at each step.

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7. Braids

7.4. Artin presentation of the braid group The braid group 𝐵𝑛 was discovered by Emil Artin in 1925 as a geometric object, but he soon obtained its purely algebraic interpretation by writing out its presentation. Theorem 7.2. The geometric braid group 𝐵𝑛 has the following presentation: ⟨𝑏1 , . . . , 𝑏𝑛−1 ∣ 𝑏𝑖 𝑏𝑖+1 𝑏𝑖 = 𝑏𝑖+1 𝑏𝑖 𝑏𝑖+1 , 𝑖 < 𝑛 − 1, 𝑏𝑖 𝑏𝑗 = 𝑏𝑗 𝑏𝑖 , |𝑖 − 𝑗| ≥ 2⟩. About the proof. It is easy to show that the generators 𝑏𝑖 of 𝐵𝑛 satisfy the relations indicated in the presentation — one must simply take a good look at Figure 7.5. The fact that these relations suffice to determine 𝐵𝑛 is not at all obvious: we omit its proof. □ i

i+1 j j+1

i

i+1 j j+1

i

i+1 i+2

=

i

i+1 i+2

=

Figure 7.5. Geometry of the braid relations

Further algebraic study of the braid group is outside the scope of this course. Here we only mention that the braid group has important applications in topology, complex analysis, theoretical physics, but it interests us because there is a simple construction, called closure (see Section 7.6 below), that assigns a knot or link to any braid. But before going on to this, we digress on algorithmically undecidable problems in group theory and knot theory.

7.5. Digression on undecidable problems

63

7.5. Digression on undecidable problems In this section, I will digress about decidable and undecidable problems in group theory (in particular in the braid group) and in knot theory. These topics will not be studied in the course, there will be no proofs, but in studying knots, links, and braids, it is necessary to know what problems in the theory are solvable in principle and what problems are undecidable (cannot be solved in principle). Also, the undecidability of many fundamental problems of group theory, which are not related to knot theory in any way, is part of the basic mathematical culture that students must acquire sooner or later — the sooner the better, in my opinion. We begin with some bad news. Let 𝐺 be a group presented as ⟨𝑔1 , . . . , 𝑔𝑛 ∣ 𝑅1 = . . . = 𝑅𝑘 = 𝑒⟩. Then the word problem in 𝐺 is as follows: Does there exist an algorithm −1 which, given two words in the alphabet {𝑔1 , 𝑔−1 1 , . . . , 𝑔𝑛 , 𝑔𝑛 }, tells us whether these two words represent the same element of 𝐺? The first bad news is Fact 1. There exist groups for which the word problem is undecidable, and most (in a certain natural sense) groups have this property. Fortunately for us, here is some good news: Fact 2. The word problem in the braid group 𝐵𝑛 is decidable for any 𝑛. The conjugation problem in 𝐺 is as follows: Does there exist an algo−1 rithm which, given two words 𝑤 1 , 𝑤 2 in the alphabet {𝑔1 , 𝑔−1 1 , . . . , 𝑔𝑛 , 𝑔𝑛 }, tells us whether these two words are conjugate in 𝐺, i.e., whether there exists a word 𝑤 such that 𝑤𝑤 1 𝑤−1 = 𝑤 2 ? The bad news here is Fact 3. There exist groups for which the conjugation problem is undecidable, and most (in a certain natural sense) groups have this property. But we have the following good news: Fact 4. The conjugation problem in the braid group 𝐵𝑛 is decidable for any 𝑛. However,

64

7. Braids

Fact 5. There exists no algorithm which, given any group presentation, correctly answers the following questions: Is the given group trivial? Is it Abelian? Is it cyclic? Is it solvable? As well as many other questions of that type. We conclude this digression by looking at algorithmic decidability in knot theory. Concerning the main problem of knot theory, we have Fact 6. There exists an algorithm which, given two knot diagrams, correctly tells us whether or not they represent the same knot. In particular, this means that there exists an algorithm which, given a knot diagram, correctly tells us whether it represents the unknot. Unfortunately, Fact 6 is a practically useless “existence theorem” — the existence of the required algorithm is rigorously proved, but the actual algorithm is not sufficiently well described to implement it as a program for humans and/or computers.

7.6. Closure of a braid The closure operation for braids is defined as shown in Figure 7.6. In the figure, we see that the closure of a braid can be an oriented knot (as in Figure 7.6(a)) or an oriented link (as in Figure 7.6(b)). The closure of a braid 𝑏 is denoted by cl(𝑏). In fact, any oriented link can be obtained as the closure of an appropriate braid, as the following theorem, due to J.W. Alexander, tells us. Theorem 7.3. For any oriented link 𝐿, there exists a braid 𝑏 such that cl(𝑏) = 𝐿. Proof. We first illustrate the main ideas of the proof by Figure 7.7, which shows how to find a braid whose closure is the eight knot 41 . This is done in two steps. In the first one (Figure 7.7(b)), we isotope the given knot into a circular one, i.e., an oriented knot whose tangent vector is always directed to the left when we look at it from a fixed point, called the center. In the second step, we “unroll” the circular knot into a closed braid (Figure 7.7(c)). In the general case of an arbitrary knot 𝐾, it suffices to prove that we can isotope it into a circular one, because the unrolling procedure

7.6. Closure of a braid

65

(a) (b) Figure 7.6. Closures of braids

C

C

(a)

(b)

C

(c)

Figure 7.7. Finding a braid whose closure is the eight knot

is the same as in the particular case of the eight knot. To perform this isotopy, we first choose the center point 𝐶 somewhere in the middle of the diagram and paint in some color, say red, all the edges of the knot that are oriented in the wrong direction (i.e., point to the right, instead of the left, if observed from 𝐶). Let [𝐴, 𝐵] be such a colored edge. Without loss of generality, we can assume that no more than one edge of 𝐾 crosses [𝐴, 𝐵], because if there is more than one, we simply subdivide [𝐴, 𝐵] into smaller edges, each crossed by only one other edge of 𝐾.

66

7. Braids

We consider two cases. In the first case, the edge that crosses [𝐴, 𝐵] forms an underpass. We then choose a point 𝑂 “behind and above” 𝐶 (see Figure 7.8(a)) and perform the Δ-move replacing [𝐴, 𝐵] by [𝐴, 𝐶] ∪ [𝐶, 𝐵]; if 𝐶 is high enough, this will indeed be a legal Δ-move. In the second case, when the edge that crosses [𝐴, 𝐵] forms an overpass, we choose the point 𝑂 “behind and below” 𝐶 and perform the same Δ-move. Note that no new red edges arise in these constructions, so that we can successively get rid of all the red edges, obtaining a circular knot. This proves the theorem for arbitrary knots. The proof for arbitrary links is similar and left to the reader (Exercise 7.14). □

7.7. Exercises 7.1. Show that the group 𝐵3 is not Abelian. 7.2. Express the braids shown in Figure 7.1 in terms of the canonical braid generators. 7.3. Are the two 4-strand braids on the left-hand side of Figure 7.2 isotopic? 7.4. Prove that the group presented as ⟨𝑔 ∣ ⟩ (the free group in one generator) is isomorphic to ℤ (the additive group of integers). 7.5. Prove that a group whose generators pairwise commute (i.e., 𝑔ℎ𝑔−1 ℎ−1 = 𝑒 for any pair of generators 𝑔, ℎ) is Abelian. 7.6. Find a presentation of the permutation group of 𝑛 objects 𝑆𝑛 . B

B

A

B

A O (a)

A

A

O C

B

O C

(b)

O C

Figure 7.8. Proof of the Alexander theorem

C

7.7. Exercises

67

7.7. Prove that the group with two generators 𝑎, 𝑏 and unique relation 𝑎𝑏𝑎−1 𝑏−1 = 1 is isomorphic to ℤ ⊕ ℤ. 7.8. Construct an epimorphism of the braid group 𝐵𝑛 onto the permutation group on 𝑛 − 1 elements. 7.9. Find a braid whose closure is the 71 knot. 7.10. Find a braid whose closure is the 52 knot. 7.11. * A braid is called pure if the endpoints of each strand have the same 𝑥, 𝑦 coordinates. Prove that pure braids on 𝑛 strands form a subgroup of 𝐵𝑛 denoted 𝑃𝐵𝑛 and find a minimal set of generators for 𝑃𝐵𝑛 . 7.12. Show that the conjugate of a given braid by another braid has the same closure as the given braid, i.e., for all braids 𝑏, 𝑏0 ∈ 𝐵𝑛 we have cl(𝑏−1 𝑏0 𝑏) = cl(𝑏). 7.13. If a braid 𝑏 ∈ 𝐵𝑛 can be expressed in terms of the generators ±1 𝑏1 , . . . , 𝑏𝑛−2 , then cl(𝑏𝑏𝑛−1 ) = cl(𝑏). 7.14. Prove the Alexander theorem for links. 7.15. Find a presentation of the braid group 𝐵3 (only one relation suffices). * For those who know how to use the graphics software from Wolfram’s “Mathematica”.

Lecture 8

Discriminants and Finite Type Invariants

In this lecture, we describe and illustrate, by means of examples, a general method due to Thom, Arnold, and Vassiliev for defining invariants of various geometric and topological objects. On a very rough and intuitive level, the method consists in studying nice (i.e., without singularities) objects (e.g. knots and links) all together with their singular versions (e.g. singular knots and links, i.e., closed curves in space with self intersections) and, using the singular objects (which form the discriminant of the theory), to define and compute invariants. Such invariants are said to be of finite type. Actually, many classical invariants discovered before the Thom–Arnold–Vassiliev discriminant theory was developed turned out to be invariants of finite type. In this lecture, we will describe four such invariants.

8.1. Discriminant of quadratic equations and real roots Given a quadratic equation 𝑥2 + 𝑝𝑥 + 𝑞 = 0, let us to find out how many distinct real roots it has. Of course, we know that it has two real roots if its discriminant Δ = 𝑝2 − 4𝑞 is positive, one if Δ is zero, and none if it is negative. But here, in order to illustrate the discriminant method, we will describe a geometric algorithm for finding the number of real roots 𝑅(𝑝, 𝑞) of the quadratic equation 𝑥2 + 𝑝𝑥 + 𝑞 = 0. 69

70

8. Discriminants and Finite Type Invariants In the (𝑝, 𝑞) coordinate plane ℝ2 , let us picture the parabola 𝑃 = {(𝑝, 𝑞) ∣ 𝑞 = 𝑝2 /4}.

Each point 𝑝, 𝑞 ∈ ℝ2 represents a specific quadratic equation. Let a concrete point (𝑝1 , 𝑞1 ) be given. How many roots does the equation 𝑥2 + 𝑝1 𝑥 + 𝑞1 = 0 have? To find this out, we proceed as follows. We start at the point (0, 1) (where 𝑅(0, 1) = 2) and join the point (𝑝1 , 𝑞1 ) by a smooth path in general position w.r.t. the parabola 𝑃. At all the points of the path up to its first intersection with the parabola (the point 𝐴 in Figure 8.1), we put 𝑅 = 2, and after crossing the parabola, 𝑅 = 0, until the next intersection (the point 𝐵 in the figure), after which the value of 𝑅 jumps to +2 and stays equal to 2 when we arrive at the point (𝑝1 , 𝑞1 ). q (p1 , q1 )

2

1 B

C D

A

E

p

0

(p1 , q1 )

0

Figure 8.1. The discriminant of quadratic equations

Figure 8.1 also shows the process of finding 𝑅 at another point, (𝑝1′ , 𝑞′1 ), of the plane, obtained by joining (0, 1) to this point by a curve that intersects the parabola at the points 𝐶, 𝐷, 𝐸. In this case the number of roots is 0. Note that 𝑅 is an invariant of the two open subsets of ℝ2 into which the plane is divided by the parabola 𝑃. Note also that when we cross the parabola, the value of 𝑅 changes according to the following rule, that

8.2. Degree of a point w.r.t. a curve

71

we call the discriminant relation: approaching the parabola, look at its orientation, if its is oriented to the left (as at the point 𝐴 on the figure), subtract 2 from the current value of 𝑅, if it is oriented to the right, we add 2 to the current value. (The reader may want to ask why we should need such a complicated algorithm for finding 𝑅, when to do that it suffices to find the sign of Δ? The answer is — because of its generality: similar geometric algorithms work in many completely different situations, when no computational algorithms are available). Theorem 8.1. The value of 𝑅(𝑝1 , 𝑞1 ) does not depend on the choice of path joining the initial point to (𝑝1 , 𝑞1 ), so that the process described above gives the correct value of the number of roots of any quadratic equation. The (rather obvious) proof is the object of Exercise 8.1.

8.2. Degree of a point w.r.t. a curve Consider a smooth immersed oriented curve 𝐶 in the plane (“immersed” means that the curve has a finite number of singularities, all of which are transversal self-intersections). Let 𝑂 be a point outside a 2-disk containing 𝐶 and 𝐴 be a point not contained in 𝐶. Then we define the degree of the point 𝐴 w.r.t. 𝐶, denoted deg(𝐴 ∶ 𝐶), as the integer obtained by the following process. Construct a smooth path 𝛼 in general position w.r.t. 𝐶 joining 𝑂 to 𝐴. At all the points on the path up to the first intersection with 𝐶 (the point 𝑋 in Figure 8.2), put deg = 0. At all the points on the path from 𝐴 up to the next intersection with 𝐶 (the point 𝑌 in Figure 8.2) change deg to deg ±1, depending on the type of crossing: if the branch of the curve which is intersected is directed to our right, we add 1 to the degree, if it directed to our left, we subtract 1. Next, when we cross 𝐶 for the second time at 𝑌 , we add ±1 according to the same rule, which may be described by the following discriminant relation (which resembles a skein relation!). deg(

) + deg(

) = deg(

) = 1.

We continue in the same way until we arrive at 𝐴, setting deg(𝐴 ∶ 𝐶) equal to the integer obtained after the last crossing.

72

8. Discriminants and Finite Type Invariants

0

1

2

0

2 1

1 1

−1

1

A

2

2

1

1 Y

X O

Figure 8.2. Finding the degree of 𝐴 w.r.t. 𝐶

But is deg(𝐴 ∶ 𝐶) well defined, doesn’t it depend on the choice of the path 𝛼? Experiments with Figure 8.2, where another path joining 𝑂 to 𝐴 is shown and the value of the degree in each connected component of ℝ2 ⧵ 𝐶 is indicated, confirm the conjecture that deg(𝐴 ∶ 𝐶) is indeed well defined. But we need a proof. Theorem 8.2. The value of deg(𝐴 ∶ 𝐶) does not depend on the choice of path joining the initial point 𝑂 to 𝐴, so that the process described above gives the correct value of the degree of a point w.r.t. a curve. The (nontrivial!) proof is the object of Exercise 8.4.

8.3. Inertia index of a quadratic form Recall that a quadratic form 𝑄 = 𝑎11 𝑥2 + 2𝑎12 𝑥𝑦 + 𝑎22 𝑦2 given by its matrix 𝑎 𝑎12 (𝑄) = ( 11 ) 𝑎21 𝑎22 can be diagonalized by a change of variable via an invertible matrix to 2 2 read 𝑄 = ±𝑥 ± 𝑦 . The number of plus signs in the last expression will

8.3. Inertia index of a quadratic form

73

be denoted by 𝜃 and called the (positive) inertia index of the quadratic form 𝑄. Sylvester’s law of inertia asserts that 𝜃 depends only on 𝑄. The discriminant Δ(𝑄) is the double infinite cone 2 Δ(𝑄) = {𝑎11 , 𝑎22 , 𝑎12 ∣ 𝑎11 𝑎22 − 𝑎12 = 0}

in 3-space with coordinates (𝑎11 , 𝑎22 , 𝑎12 ). It is shown in Figure 8.3. Let us orient the right part of the cone so that its orientation is counterclockwise (looking at it from the outside) and the left part of the cone so that its orientation is clockwise (also looking at it from the outside). a12

1 a11

P0

0

A B

C

2 P1

a22 Figure 8.3. Discriminant of a quadratic form

Then the value of 𝜃 will be +2 inside the right part of the cone, to 0 inside the left part of the cone, and to +1 outside the double cone. Starting at the point 𝑃0 = (0, 0, 1) (where 𝜃 = +1), we can find the value of the inertia index 𝜃 at any other point 𝑃1 = (𝑋1 , 𝑦1 , 𝑧1 ) by the following procedure. We join 𝑃0 to 𝑃1 by a smooth path in general position w.r.t. the cone and move along the path until the first intersection with the cone (the point 𝐴 in Figure 8.3). If the orientation of the cone appears to be counterclockwise as we approach 𝐴 (as in the figure), we add 1 to the current value of 𝜃 (equal to +1) to obtain the value of 𝜃 at points near 𝐴 on the other side of the cone, while if the orientation appears clockwise, we subtract 1 from 𝜃. After that, we proceed along the path to the next intersection with the cone (the point 𝐵 in the figure), where we will have

74

8. Discriminants and Finite Type Invariants

to subtract 1 from 𝜃, and so on until we reach 𝑃1 . The current value of 𝜃 as we approach 𝑃1 (0 in the figure) will be equal to the inertia index of the quadratic form determined by the coordinates of 𝑃1 . Theorem 8.3. The result of the process described above (𝜃(𝑃1 )) does not depend on the choice of the path joining 𝑃0 to 𝑃1 . The (rather simple) proof of this theorem is the object of Exercise 8.5.

8.4. Gauss linking number The Gauss linking number of a pair of closed curves in ℝ3 indicates the number of times one closed curve winds around the other (see Figure 8.4, where the linking numbers are 3, 2, −3, −1, respectively).

(A)

(B)

(C)

(D)

Figure 8.4. Gauss linking number of two-component links

Consider a two-component link diagram 𝐿 = 𝐿1 ∪ 𝐿2 . First assume that 𝐿1 = ○ is the round circle. Choose an arbitrary point 𝑃0 ∈ 𝐿2 outside ○ and move along 𝐿2 , crossing the circle ○ successively at the points 𝑃1 , 𝑃2 , . . . , 𝑃2𝑘−1 , 𝑃2𝑘 . Put 𝑘

lk(○, 𝐿2 ) = ∑ 𝜀𝑖 , 𝑖=1

where 𝜀𝑖 = 0 if the signs of the crossings at the points 𝑃2𝑖−1 and 𝑃2𝑖 are opposite, 𝜀𝑖 = +1 (resp. −1) if the signs of the crossings at 𝑃2𝑖−1 and 𝑃2𝑖 are both positive (resp. both negative). Now consider the general case, when both components 𝐿1 and 𝐿2 of the link are arbitrary. Denote by 𝑛1 (resp. 𝑛3 ) the number of crossings where 𝐿1 overpasses 𝐿2 and forms a positive (resp. negative) crossing and

8.4. Gauss linking number

75

denote 𝑛2 (resp. 𝑛4 ) the number of crossings where 𝐿2 overpasses 𝐿1 and forms a positive (resp. negative) crossing. Then lk(𝐿1 , 𝐿2 ) = 𝑛1 − 𝑛4 = 𝑛2 − 𝑛3 . Theorem 8.4. The above definition yields an integer, called Gauss linking number, which is an isotopy invariant of the two-component link 𝐿 = 𝐿1 ∪ 𝐿2 . We shall not prove this theorem here. There are two different proofs, one that goes back to Gauss and based on the concrete definition of lk(𝐿1 , 𝐿2 ) as the double integral lk(𝐿1 , 𝐿2 ) =

det(𝛼1̇ (𝑠), 𝛼2̇ (𝑡), 𝛼1 (𝑠) − 𝛼2 (𝑡)) 1 ∫ 𝑑𝑠 𝑑𝑡, 4𝜋 𝕊1 ×𝕊1 |𝛼1 (𝑠) − 𝛼2 (𝑡)|3

where 𝛼1 , 𝛼2 ∶ 𝕊1 → ℝ3 are parametrisations of 𝐿1 and 𝐿2 , and the dot denotes differentiation. The second proof is based on homology theory and defines lk as the intersection number of 𝐿1 with a surface spanning 𝐿2 . Remark 8.1. Let us explain what lies behind this construction and, in particular, describe the discriminant underlying the Gauss linking number. In the previous examples, the objects under study (quadratic equations, points in ℝ2 ⧵ 𝐶, quadratic forms) were represented by points in two- or 3-dimensional space and the corresponding invariants (number of real roots, degree w.r.t. 𝐶, inertia index) were defined and calculated with the help of the corresponding discriminant (the parabola 𝑃, the curve 𝐶, the double cone Δ(𝑄)). For the linking number lk(𝐿1 , 𝐿2 ), the objects (two-component links) can be represented by points in infinite-dimensional linear space, and the discriminant will be a subset of this space, also of infinite dimension. It is of course impossible to draw a realistic picture of the discriminant, but we can give a good visual idea of what happens when we cross the discriminant because its codimension is equal to one. This is done by a discriminant relation that may be called the Gauss skein relation: lk(

) − lk(

) = lk(

) = 1.

Note that all the other discriminants were of the same codimension (equal to 1). We can say that all four examples are finite type invariants

76

8. Discriminants and Finite Type Invariants

of degree one, because they are determined by codimension 1 discriminants.

8.5. Exercises 8.1. Prove Theorem 8.1. 8.2. * For third degree equations 𝑥3 + 𝑝𝑥2 + 𝑞𝑥 + 𝑟 = 0, represent the discriminant Δ = 𝑝2 𝑞2 − 4𝑞3 − 4𝑝3 𝑟 + 18𝑝𝑞𝑟 − 27𝑟2 in (𝑝, 𝑞, 𝑟)-space and indicate various discriminant relations used to find the number of real roots of the equation. 8.3. In Figure 8.2, the values of the degree of points in the connected components of ℝ2 ⧵ 𝐶 are shown, but one of them is incorrect. Find it and correct it. 8.4. * Prove Theorem 8.2. 8.5. Prove Theorem 8.3. 8.6. What is the linking number of the left Hopf link and of the right Hopf link? 8.7. * If the linking number of a two-component link diagram is zero, does it follow that the components can be separated (i.e., the diagram is equivalent to a link diagram whose components lie in different half planes)? 8.8. Verify that the linking number of each pair of curves in Figure 8.4 is indicated correctly by calculating the linking numbers according to the procedure described in Section 8.4. * For those who know how to use the graphics software from Wolfram’s “Mathematica”.

Lecture 9

Vassiliev Invariants

In this lecture, we begin the study of Vassiliev invariants, also known as the Goussarov–Vassiliev invariants or finite-type knot invariants. There is an infinite series of Vassiliev invariants, each one is a real-valued (or complex-valued, if we prefer) function 𝑣 defined on the set of oriented knots. Vassiliev’s construction of his knot invariants is particular case of the general construction, called the Thom–Arnold–Vassiliev discriminant method and described in the previous lecture. The main idea of the method is the following. In order to study the given nice generic objects (say knots, i.e., smooth embeddings 𝕊1 → ℝ3 ) we study, simultaneously with the nice objects, their singular analogues (singular knots, i.e., smooth maps 𝕊1 → ℝ3 ). The set 𝑆 of all singular knots has a natural infinite-dimensional linear space structure, in which knots form an everywhere dense open subset 𝒦; the set Δ ≔ 𝑆 ⧵ 𝒦 is the discriminant of the system; it is stratified as Δ = Δ1 ∪Δ2 ∪. . ., where Δ1 is the set of singular knots with one transversal self-intersection, Δ2 is the set of singular knots with two transversal self-intersections, and so on. The set Δ1 is of codimension 1 in the linear space 𝑆 and divides 𝑆 into compartments, in which the all points (knots) have the same values of any given Vassiliev invariant 𝑣. When we move from a compartment to an adjacent one, the value of 𝑣 undergoes a jump determined by the “Vassiliev skein relation”. For the details, see below.

77

78

9. Vassiliev Invariants

We shall introduce the Vassiliev invariants axiomatically, assuming that functions satisfying the given axioms exist. We shall learn how to compute them in simple cases and study their properties. There are two different ways of proving the existence of Vassiliev invariants. The first, Vassiliev’s original proof based on the Vassiliev cohomology spectral sequence, lies outside the framework of this course. The second is based on the Kontsevich integral, which we shall discuss in Lecture 11. It is conjectured that Vassiliev invariants are complete, i.e.,two knots 𝑘1 , 𝑘2 are not ambient isotopic if and only if there exists a Vassiliev invariant 𝑣 ∶ 𝒦 → ℝ such that 𝑣(𝑘1 ) ≠ 𝑣(𝑘2 ). It can be shown that Vassiliev invariants are stronger than the Jones polynomial: the coefficients at various powers of 𝑞 of the Jones polynomial of a knot can be recovered from the values of certain Vassiliev invariants of that knot.

9.1. Basic definitions Denote by 𝒦 the set of oriented knots, i.e., smooth embeddings 𝑘 ∶ 𝕊1 → ℝ3 . By Σ denote the set of immersions 𝑖 ∶ 𝕊1 → ℝ3 , i.e., smooth maps whose singularities consist of a finite number of transversal self-intersections. Further, write Σ0 ≔ 𝒦 and denote by Σ1 the set of all singular knots with one transversal self-intersection, by Σ2 , the set of all singular knots with two transversal self-intersections, and so on. We have the following infinite sequence ∞

𝒦 = Σ 0 ⊂ Σ 1 ⊂ . . . ⊂ Σ𝑛 ⊂ . . . ⊂ Σ =

⋃

Σ𝑛 .

𝑛=0

Elements of Σ𝑛 for 𝑛 ≥ 1 are called singular knots with 𝑛 double points, the set Δ = ⋃𝑛≥1 Σ𝑛 is the discriminant, the set Σ1 is the generic part of the discriminant Δ. To illustrate the situation, we assume without proof that the set 𝒞 of all smooth maps 𝕊1 → ℝ3 has a natural infinite-dimensional linear space structure, the closure of 𝒦 is the whole space 𝒞, the closure Σ1 of Σ1 divides 𝒞 into compartments. All points of each compartment are knots ambient isotopic to each other and the generic part of the discriminant Σ1 in a neighborhood of each of its points is a hypersurface of codimension 1 in the linear space 𝒞. We choose a coorientation on Σ1 by

9.1. Basic definitions

79

specifying a nonzero vector transversal to Σ1 at each point if Σ1 so that this vector changes continuously when we move along Σ1 . Suppose that two knots 𝑘+ , 𝑘− living in adjacent compartments are joined by a path 𝛼 that transversally intersects the generic part Σ1 of Δ between the compartments at a point 𝑘0 . This point represents a singular knot with exactly one double point. Then as we move along 𝛼, a crossing change at the point corresponding to the double point of 𝑘0 occurs, and 𝑘+ is transformed into 𝑘− . This is (very schematically!) illustrated in Figure 9.1, where 𝑘+ is a trefoil that becomes an unknot after going through Σ1 . The knots Σ1 Σ2

k+

k− k0

k+

k−

k0

C Figure 9.1. The crossing change that occurs when traversing Σ1

on Σ1 are all singular with one double point, as is the knot 𝑘0 . In the figure, Σ1 is represented as a graph in the plane, although actually it is an infinite-dimensional (nonlinear!) subspace of codimension one in an infinite-dimensional linear space. The self-intersections of Σ1 are shown as points, although actually they are also nonlinear infinite-dimensional subsets Σ2 (of codimension two), their elements being singular knots with two double points. The “deeper parts” of the discriminant (Σ𝑘 , 𝑘 ≥ 3) are not represented by anything in the figure, which is very schematical and gives no idea of the complexity of the overall picture. A Vassiliev invariant is defined as a function 𝑣 ∶ Σ → ℝ that satisfies the Vassiliev skein relation 𝑣(𝑘+ ) − 𝑣(𝑘− ) = 𝑣(𝑘∘ )  v

[



 =v

X



 −v

U

 ,

80

9. Vassiliev Invariants

where 𝑘∘ is an arbitrary singular knot with 𝑛 + 1 double points (𝑛 ≥ 0), 𝑘+ and 𝑘− are singular knots with 𝑛 double points obtained by desingularizing 𝑘∘ as shown in the figure. We say that 𝑣 is a Vassiliev invariant of order ≤ 𝑛 if it satisfies the finite-type condition 𝑣|Σ = 0 𝑠

for any 𝑠 > 𝑛.

9.2. The one-term and four-term relations The set 𝑉𝑛 of all Vassiliev invariants of order ≤ 𝑛 has a linear space structure (𝑣 1 + 𝑣 2 )(𝑘) = 𝑣 1 (𝑘) + 𝑣 2 (𝑘),

(𝜆𝑣)(𝑘) = 𝜆 ⋅ 𝑣(𝑘).

We have the following sequence of inclusions of linear spaces 𝑉0 ⊂ 𝑉1 ⊂ . . . ⊂ 𝑉𝑛 ⊂ . . . Vassiliev invariants have the following properties . (I) One-term relation:

S

 v

 = 0.

This easily follows from the Vassiliev skein relation if we assume (which we do!) that any 𝑣 ∈ 𝑉𝑛 , 𝑛 ≥ 0 is an ambient isotopy invariant, but it is more convenient for us to prove a slightly more general statement, namely   v = 0.

=

This is done by means of a lovely trick (see Exercise 9.1). (II) Four-term relation:  v

@



 −v

?



6

 +v



5

 −v

 = 0.

9.4. Chord diagrams

81

To prove this, we apply the skein relation four times (using four different double points)       v =v −v = a − b,  v  v  v

@ ? 6 5



 =v



 =v



 =v

7 ; : ;



 −v



 −v



 −v

= > 7 8

 = c − d,  = c − a,  = d − b,

where the letters 𝑎, 𝑏, 𝑐, 𝑑 denote the value of the given Vassiliev invariant 𝑣 on identical ambient isotopic singular knots. Adding these four equalities, we obtain (𝑎 − 𝑏) − (𝑐 − 𝑑) + (𝑐 − 𝑎) − (𝑑 − 𝑏) = 0 as claimed.

9.3. Dimensions of the spaces 𝑉𝑛 There is an infinite number of Vassiliev invariants. We shall learn how to compute the value of some of them (of small order) for concrete knots a little later, but first let us try to answer the following natural question: How “big” is the space 𝑉𝑛 ? Here we shall characterize its “size” by its dimension, which has been computed for 𝑛 ≤ 12. We present the dimensions of the linear spaces of Vassiliev invariants of order strictly equal to 𝑛, i.e., 𝒱𝑛 ≔ 𝑉𝑛 /𝑉𝑛−1 , for 𝑛 = 0, 1, 2, . . . , 12: dim 𝒱𝑛 = 1, 0, 1, 1, 3, 4, 9, 14, 27, 44, 80, 132, 232. The proof for large values of 𝑛 is beyond the scope of this course, but we will find the dimensions of 𝒱𝑛 for 𝑛 = 0, 1, 2, 3, 4. To do this, we will need the fundamental notion of chord diagram.

9.4. Chord diagrams To each singular knot 𝑘, we associate a chord diagram as follows: the knot 𝑘 is an immersion of the circle 𝕊1 into ℝ3 ; let us number the double

82

9. Vassiliev Invariants

points of 𝑘 in the order of their appearance as we go around the curve 𝑘, mark their two preimage points on 𝕊1 by the same number, and join each pair of identical numbers by chords; the obtained figure, denoted 𝐷(𝑘), is called the chord diagram of the singular knot 𝑘. An example of this construction is shown in Figure 9.2. 1 1=4

2 6 2=5

3=6

5

3 4

Figure 9.2. Example of a chord diagram

Two chord diagrams are considered identical if there is a bijection between the chords preserving the order of their endpoints around the circle. Thus a chord diagram of a singular knot is well defined, it depends only on the knot and does not depend on the choice of double point from which we began our numbering. By definition, the chord diagram of any classical (i.e., nonsingular) knot is a circle without any chords. We shall need the following lemma. Crossing Change Lemma 9.1. The value 𝑣(𝑘) of a Vassiliev invariant of order 𝑛 of a singular knot 𝑘 with 𝑛 crossings does not change under crossing changes. Proof. Since the order of 𝑣 is equal to the number of double points of 𝑘, the finite type condition and the skein relation together tell us that a crossing change “costs nothing”. □ It follows from Lemma 9.1 that a Vassiliev invariant of order 𝑛 of a singular knot 𝑘 with 𝑛 crossings depends only on the chord diagram of 𝑘, and so in that situation, we shall write 𝑣(𝐷(𝑘)) (where 𝐷(𝑘) is a picture of the chord diagram) instead of 𝑣(𝑘).

9.5. Vassiliev invariants of small order

83

9.5. Vassiliev invariants of small order We obviously have dim 𝑉0 = 1, because one can pass from any knot diagram to the unknot by a succession of crossing changes (Trivialization Lemma 2.1), and these crossing changes “cost nothing”, because in the particular case 𝑛 = 0, the skein relation reads       v =v −v ,

[

X

U

where the singular knot on the left-hand side of the equation has exactly one double point (the one inside the circle) and the two knots in the right-hand side have no singularities. Therefore, we can set 𝑣(○) equal to any real number, and it will satisfy the axioms of the theory. Thus dim 𝑉0 = dim 𝒱0 = 1. Now let 𝑛 = 1. It turns out that we have dim 𝒱1 = dim(𝑉1 /𝑉0 ) = 0, which means that 𝑛 = 1 gives us no new invariants: all Vassiliev invariants of order 1 are actually of order 0. The proof of this disappointing result is the object of Exercise 9.4. Fortunately, when we pass to 𝑛 = 2, we obtain significant results. Denote by 𝑣 2 the Vassiliev invariant of order 2 such that 𝑣(

) = 1,

𝑣 2 (○) = 0,

where denotes the chord diagram with two intersecting chords, and also assume that 𝑣 2 vanishes on the chord diagram with two nonintersecting chords. Let us compute the value of 𝑣 2 on the right trefoil 𝑘. We have: 𝑣2 (

) = 𝑣2 ( = 𝑣2 (

) + 𝑣2 ( ) = 𝑣2 (

) ) + 𝑣2 (

) = 𝑣2 (

)

and, since the last pictured knot has a chord diagram with two intersecting chords, we have 𝑣 2 (𝑘) = 1. This is a meaningful result, since it implies that the trefoil is not the trivial knot.

84

9. Vassiliev Invariants

It is easy to show (Exercise 9.5) that the value of 𝑣 2 on the left trefoil is the same as that on the right one. However, Vassiliev invariants of order three already distinguish the right trefoil from the left one. It suffices to choose a Vassiliev invariant 𝑣 3 whose value on the chord diagram consisting of three pairwise intersecting chords is equal to 1. The computation that follows shows that the values of 𝑣 3 on the left and right trefoil differ by 1, so that the two trefoils are not isotopic: ) − 𝑣3 (

𝑣3 (

) = 𝑣3 (

) − 𝑣3 (

) = 𝑣3 (

).

Now let us compute the value of 𝑣 2 of the 51 knot. We obtain 𝑣2 (

) = 𝑣2 ( = 𝑣2 (

) − 𝑣2 (

) − 𝑣2 (

)

) − 𝑣2 ( = 𝑣2 (

) − 𝑣2 ( ) − 2𝑣 2 (

)

)

and therefore 𝑣2 (

) = 2𝑣 2 (

) + 𝑣2 (

) = 3.

This shows that the knot 51 is not isotopic to the trefoil. It also shows, together with the calculation in Exercise 9.6, that the knot 51 is not isotopic to the eight knot 41 . We are not going to learn how to compute the values of high order Vassiliev invariants. These computations, which involve the inductive construction of the so-called actuality tables, are rather cumbersome.

9.6. Exercises 9.1. Prove the one-term relation in its more general form.

9.6. Exercises 9.2. Draw the chord diagrams of the following singular knots.

9.3. Find singular knots with the following chord diagrams.

9.4. Prove that dim 𝒱1 = 0. 9.5. Calculate 𝑣 2 for the left trefoil. 9.6. Calculate 𝑣 2 for the eight knot. 9.7. Calculate 𝑣 2 for the 52 knot. 9.8. Prove that dim 𝒱2 = 1.

85

Lecture 10

Combinatorial Description of Vassiliev Invariants

In the previous lecture, we defined Vassiliev invariants and learned how to compute their values for concrete knots in the simplest cases. To do this in more complicated situations, an additional rather intricate tool, called “actuality tables” (see [CD], p. 74) is required. We will not study actuality tables, because our primary interest is not the practical computation of concrete values of 𝑣(𝑘) ∈ ℂ, 𝑣 ∈ 𝑉𝑛 , 𝑘 ∈ 𝒦, but the study of the spaces 𝑉𝑛 themselves. We shall see that they have a rich algebraic structure and possess a beautiful combinatorial description in terms of chord diagrams. We begin with some algebraic preliminaries. .

10.1. Digression: graded algebras By definition, an algebra 𝐴 over ℂ is a linear space (over ℂ) with a commutative associative operator (the product) ⋅ ∶ 𝐴 × 𝐴 → 𝐴 such that (𝛼𝑥 + 𝛽𝑦) ⋅ (𝛾𝑧) = 𝛼𝛾(𝑥 ⋅ 𝑧) + 𝛽𝛾(𝑦 ⋅ 𝑧)

∀𝛼, 𝛽, 𝛾 ∈ ℂ, ∀𝑥, 𝑦, 𝑧 ∈ 𝐴.

A graded algebra 𝐴 over ℂ is an algebra presented as the infinite sum of linear subspaces 𝐴 = 𝐴1 ⊕ 𝐴2 ⊕ . . . ⊕ 𝐴𝑛 ⊕ . . . satisfying the condition 𝐴𝑝 ⋅ 𝐴𝑞 ⊂ 𝐴𝑝+𝑞 . The subscript 𝑛 is the grading. 87

88

10. Combinatorial Description of Vassiliev Invariants

The simplest nontrivial example of a graded algebra is the algebra of polynomials ℂ[𝑥] with complex (or real) coefficients; for it, the product is the ordinary product of polynomials and the grading is the degree of the polynomial. If the commutativity property of the algebra is replaced by skew commutativity, 𝑥𝑦 = (−1)𝑝𝑞 𝑦𝑥, 𝑥 ∈ 𝐴𝑝 , 𝑦 ∈ 𝐴𝑞 , then one obtains a skew-commutative algebra, important examples of which are differential forms and cohomology groups.

10.2. The graded algebra of chord diagrams Linear combinations with coefficients in ℂ of chord diagrams with 𝑛 chords have a natural linear space structure, denoted by 𝐷𝑛 for any 𝑛 ≥ 0. Here is an example of an element of 𝐷3 : − (√2 + 𝑖)

𝐷3 ∋ 2

+ √3

.

The dimension of 𝐷3 is 5, because there are 5 different 3-chord diagrams ,

,

,

,

,

which form a basis for 𝐷3 . ∞

Let 𝐷 be the linear space 𝐷 = ⨁ 𝐷𝑛 . 𝑛=0

For any 𝑛 ≥ 2, we define the four-term relation for 𝑛-chord diagrams as −

+

−

= 0,

where only two of the 𝑛 chords are shown in each of the 𝑛-chord diagrams, the remaining 𝑛 − 2 chords (not shown) are exactly the same in all four diagrams and do not have any endpoints in the little fat arcs of the circles. The use of the name “four-term relation” is motivated by the following statement.

10.2. The graded algebra of chord diagrams

89

Lemma 10.1. If 𝑣 ∈ 𝑉𝑛 , then

𝑣(

) − 𝑣(

) + 𝑣(

) − 𝑣(

) = 0,

where only two of the 𝑛 chords are shown in the 𝑛-chord diagrams, the 𝑛 − 2 chords not shown are the same in all diagrams and do not have any endpoints in the little fat arcs of the circles. The proof is the object of Exercise 10.2. We now define the one-term relation for 𝑛-chord diagrams as

= 0,

where only one chord is shown and the other 𝑛 − 1 chords have no endpoints in the fat arc of the circle. The use of the name “one-term relation” comes from the relation of the same name for Vassiliev invariants of knots. We can now define the algebra of chord diagrams Δ as the linear ∞

space Δ = ⨁ Δ𝑛 , where each Δ𝑛 is the quotient space of 𝐷𝑛 by all possi𝑛=0

ble one-term and four-term relations for 𝑛-chord diagrams; Δ is actually a graded algebra — the multiplication operation will be defined below, but first, as an illustration, we shall study Δ3 . In Δ3 , the (equivalence classes of) the following three-chord diagrams ,

,

are zero by the one-term relation. By the four-term relation, we have

−

+

−

= 0,

90

10. Combinatorial Description of Vassiliev Invariants

the third summand is zero by the one-term relation, and so =2

,

which means that dim Δ3 = 1. We now define the product of two (equivalence classes of) chord diagrams 𝐶1 ∈ Δ𝑝 , 𝐶2 ∈ Δ𝑞 as in the example below (for 𝑝 = 3 and concrete choices of 𝐶1 and 𝐶2 ): ⋅

=

=

.

Lemma 10.2. The above product is well defined, i.e., it does not depend on the choice of representatives and of the places on the circles where the circles are glued together. Sketch of the proof. The independence of the choice of representatives is obvious. Let us prove that the product does not depend on the choice of the place where the circles are glued together. We will do this in the case of a concrete example, namely C0 = C4 ? =

=

=

.

To prove that the 5-chord diagrams 𝐶0 , 𝐶4 ∈ 𝐷5 are equivalent, we move the upper endpoint of the fat chord in 𝐶0 counterclockwise successively over the next endpoint four times, obtaining the chord diagrams 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 : , C0

, C1

, C2

, C3

. C4

Applying the four-term relation to the diagrams 𝐶0 , 𝐶1 , 𝐶2 , 𝐶3 and then to 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 , we see that 𝐶0 − 𝐶1 + 𝐶2 − 𝐶3 = 0

and

𝐶1 − 𝐶2 + 𝐶3 − 𝐶4 = 0.

10.3. The Vassiliev–Kontsevich theorem

91

Adding the last two equalities, we obtain 𝐶0 = 𝐶4 , as required. □

In the general case, the argument is similar.

Remark 10.1. The algebra of chord diagrams is actually a bialgebra: besides the multiplication defined above, it possesses a comultiplication. We will not define this additional structure (see [CD], p. 92).

10.3. The Vassiliev–Kontsevich theorem Let Δ𝑛 be the linear space of 𝑛-chord diagrams modulo the one-term and the four-term relations; we denote by Δ∗𝑛 its dual space 1 , i.e., the space of linear functions on elements of Δ𝑛 ; recall that 𝑉𝑛 denotes the linear space of Vassiliev invariants of order ≤ 𝑛. Theorem 10.1. For any 𝑛 ≥ 0, there exists an isomorphism 𝛼𝑛 ∶ 𝑉𝑛 /𝑉𝑛−1 → Δ∗𝑛 . About the proof. We will only begin the proof of this theorem, which gives a simple combinatorial description of the space of Vassiliev invariants, by constructing the map 𝛼𝑛 . Let 𝑣 𝑛 be a Vassiliev invariant of order exactly equal to 𝑛. Its image must be a linear function 𝑙 ∶ Δ𝑛 → ℂ; by linearity, it suffices to define 𝑙 on a basis 𝑑1 , . . . , 𝑑𝑠 of Δ𝑛 . We set 𝑙(𝑑𝑖 ) = 𝑣 𝑛 (𝑑𝑖 ) for all 𝑖 ∈ {1, . . . , 𝑠}, (The fact that the right-hand side of this equality is well defined easily follows from the Crossing Change Lemma 9.1.) The injectivity of 𝛼𝑛 is the object of Exercise 10.5, its surjectivity can be proved by using the Kontsevich integral, which we will study in the next lecture. □ Recall that Δ = ⨁ Δ𝑛 is a graded algebra and therefore Δ∗ = ⨁ Δ∗𝑛 𝑛≥0

𝑛≥0

inherits this structure via 𝛼−1 𝑛 . Thus, for all 𝑛 ≥ 0, the space 𝒱 = ⨁ 𝑉𝑛 /𝑉𝑛−1 of Vassiliev invariants 𝑛≥0

is a graded algebra. 1 The space ∆∗𝑛 is denoted by 𝒲𝑛 in the book [CD], where it is called the “space of unframed weight systems”.

92

10. Combinatorial Description of Vassiliev Invariants

10.4. Vassiliev invariants vs. other invariants It turns out that the space of all Vassiliev invariants is more powerful that any of the other previously known knot invariants: in fact, the previously known invariants can all be expressed in terms of Vassiliev invariants. In this section, we will state three such results without proof. Fact 10.1. The 𝑛th coefficient of the Conway polynomial is a Vassiliev invariant of order ≤ 𝑛. In the Jones polynomial of a knot, let us substitute 𝑞 = 𝑒ℎ , expand it into an infinite power series in ℎ and denote by 𝑗𝑛 the coefficient at ℎ𝑛 . Fact 10.2. The coefficient 𝑗𝑛 in the power series defined above is a Vassiliev invariant of order ≤ 𝑛. The Casson invariant 𝐶 is a classical integer-valued invariant of homology 3-spheres. (Each homology sphere is obtained from the ordinary 3-sphere by surgery along a knot 𝐾, so that 𝐶 can be regarded as a knot invariant.) Fact 10.3. The Casson invariant 𝐶 is equal to the second coefficient of the Conway polynomial and is therefore a Vassiliev invariant of order less than or equal to 2. For the proofs, see [CD].

10.5. Exercises 10.1. Find the dimensions of the linear spaces 𝐷2 and 𝐷4 . 10.2. Prove Lemma 10.1. 10.3. Find the dimension of Δ4 . 10.4. Find the dimension of Δ5 . 10.5. Prove that the map 𝛼𝑛 is injective.

10.5. Exercises

93

10.6. Prove the first two of the following relations. =

+

,

+

=2

=

+

,

=

+

+

=

+

.

10.7. Prove the last three of the relations in the picture above.

, ,

Lecture 11

The Kontsevich Integrals

The Kontsevich integrals {𝑍𝑚 } constitute a beautiful and sophisticated tool whose main purpose is to prove the existence of Vassiliev invariants by providing a proof of the Vassiliev–Kontsevich Theorem 10.1. These integrals actually form an infinite series in 𝑚 = 0, 1, 2, . . . and are assigned to any concrete knot 𝐾 ∶ 𝕊1 → ℝ3 from the class of “strictly Morse” knots. Their values are elements of the algebra Δ𝑚 of chord diagrams and their domains of definition are subsets of the Euclidean space ℝ𝑚 . In this lecture, we shall describe the Kontsevich integrals (in the form originally defined by Kontsevich) for a specific trefoil knot 𝐾𝑇𝑅 , and calculate the integral 𝑍2 (𝐾𝑇𝑅 ) in detail. We then calculate the integral of a specific unknot 𝐻 (having the shape of a hump) and learn that

𝑍2 (𝐻) ≠ 0 = 𝑍2 (○).

This means that the original Kontsevich integrals are not isotopy invariants. After that, we redefine the Kontsevich integrals, learn that their new version is isotopy invariant, and explain in what sense they provide a proof of the existence of Vassiliev invariants. 95

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11. The Kontsevich Integrals

11.1. The original Kontsevich integral of a trefoil knot Consider the trefoil knot 𝐾𝑇𝑅 ∶ 𝕊1 → ℝ3 = ℂ × ℝ shown in Figure 11.1. Note that it is a strictly Morse smooth knot, which means that it has finite number 𝑐 of extrema (here 𝑐 = 4), which are either maxima or minima and are located at different levels of the 𝑡-axis. t τ1

T τ2

z1 (t2 )

τ2

z1 (t1 )

t2

T τ2

t1

K

τ1

τ2 z1 (t1 )

τ0

z1 (t2 )

τ1 z1 (t2 ) z1 (t2 ) z1 (t1 ) z1 (t1 )

z

τ0

τ1

∈

∈

R3 = C × R z

t Figure 11.1. A strictly Morse trefoil knot

The Kontsevich integral 𝑍(𝐾𝑇𝑅 ) is defined as the series ∞

1 𝑍 (𝐾𝑇𝑅 ), 𝑚 𝑚 (2𝜋𝑖) 𝑚=0

𝑍(𝐾𝑇𝑅 ) = ∑ where

𝑚

𝑍𝑚 (𝐾𝑇𝑅 ) = ∫

∑

𝜏0 ≤𝑡1