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. 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.

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Sossinsky, A. B.;

A. B. Sosinskii

Education Development Center, Incorporated

Student mathematical library, Providence, Rhode Island, 2023

United States, United States of America

2020 Mathematics Subject Classification. Primary 55-xx, 51-xx, 20-xx.

producers:
XeTeX 0.999992

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Front Cover 1
Half Title 2
Title 4
Copyright 5
Contents 6
Foreword 12
Permissions & Acknowledgments 16
Lecture 1. Knots and Links, Reidemeister Moves 20
1.1. Main definitions 20
1.2. Reidemeister moves 23
1.3. Torus knots 27
1.4. Invertibility and chirality 27
1.5. Exercises 28
Lecture 2. The Conway Polynomial 30
2.1. Axiomatic definition 31
2.2. Calculations 31
2.3. Uniqueness and existence of the Conway polynomial 33
2.4. Chirality, orientation-reversal, and multiplicativity of the Conway polynomial 38
2.5. Exercises 38
Lecture 3. The Arithmetic of Knots 40
3.1. Boxed knots and their connected sum 40
3.2. The semigroup of boxed knots 41
3.3. Ordinary knots vs. boxed knots 43
3.4. Decomposition into prime knots 44
3.5. Some remarks about unknotting 45
3.6. Exercises 45
Lecture 4. Some Simple Knot Invariants 48
4.1. Stick number 48
4.2. Crossing number 49
4.3. Unknotting number 50
4.4. Tricolorability 51
4.5. Digression about orientable surfaces 52
4.6. Seifert surface of a knot 54
4.7. The genus of a knot 55
4.8. Exercises 55
Lecture 5. The Kauffman Bracket 58
5.1. Digression: statistical models in physics 58
5.2. The “state” of a (nonoriented) knot diagram 60
5.3. Definition and properties of the Kauffman bracket 61
5.4. Is the Kauffman bracket invariant? 62
5.5. Exercises 65
Lecture 6. The Jones Polynomial 66
6.1. Definition via the Kauffman bracket 66
6.2. Main properties of J(mskip 2mu⋅mskip 2mu) 68
6.3. Axioms for the Jones polynomial 69
6.4. Multiplicativity 70
6.5. Chirality and reversibility 71
6.6. Is the Jones polynomial a complete invariant? 71
6.7. Is V a Laurent polynomial in q? 72
6.8. Knot tables revisited 73
6.9. Exercises 74
Lecture 7. Braids 76
7.1. Geometric braids 77
7.2. The geometric braid group B_{n} 78
7.3. Digression on group presentations 79
7.4. Artin presentation of the braid group 81
7.5. Digression on undecidable problems 82
7.6. Closure of a braid 83
7.7. Exercises 85
Lecture 8. Discriminants and Finite Type Invariants 88
8.1. Discriminant of quadratic equations and real roots 88
8.2. Degree of a point w.r.t. a curve 90
8.3. Inertia index of a quadratic form 91
8.4. Gauss linking number 93
8.5. Exercises 95
Lecture 9. Vassiliev Invariants 96
9.1. Basic definitions 97
9.2. The one-term and four-term relations 99
9.3. Dimensions of the spaces V_{n} 100
9.4. Chord diagrams 100
9.5. Vassiliev invariants of small order 102
9.6. Exercises 103
Lecture 10. Combinatorial Description of Vassiliev Invariants 106
10.1. Digression: graded algebras 106
10.2. The graded algebra of chord diagrams 107
10.3. The Vassiliev–Kontsevich theorem 110
10.4. Vassiliev invariants vs. other invariants 111
10.5. Exercises 111
Lecture 11. The Kontsevich Integrals 114
11.1. The original Kontsevich integral of a trefoil knot 115
11.2. Calculation of the integral for m=2 116
11.3. Kontsevich integral of the hump 118
11.4. Results 119
11.5. Exercises 120
Lecture 12. Other Important Topics 122
12.1. Knot polynomials 122
12.2. Virtual knots 124
12.3. Knots in 3-manifolds 125
12.4. Khovanov homology 126
12.5. Knot energy 127
12.6. Connections with other fields 127
Lecture 13. A Brief History of Knot Theory 130
13.1. Carl Friedrich Gauss: pictures of knots and the linking number 131
13.2. William Thompson, P.G. Tait, J.C. Maxwell, and knots as models of atoms 132
13.3. Henri Poincaré: surgery along the trefoil and the fundamental group 133
13.4. Max Dehn, Kurt Reidemeister, the German school, and the beginnings of knot theory 134
13.5. James Alexander, John Conway, their polynomial, and the skein relation 135
13.6. Vaughan Jones, Louis Kauffman, and the discoverers of the HOMFLY polynomial 136
13.7. Edward Witten, Michael Atiyah, and quantum field theory 138
13.8. Oleg Viro, Nikolay Reshetikhin, Vladimir Turaev, and a rigorous theory of links in manifolds 139
13.9. Wolfgang Haken, Friedhelm Waldhausen, Sergei Matveev, and the classification of knots 139
13.10. Victor Vassiliev and Mikhail Goussarov, and finite type invariants 140
13.11. Maxim Kontsevich, Dror Bar-Natan, Joan Birman, and the combinatorial theory of finite type invariants 141
13.12. Concluding remarks 142
Bibliography 144
Index 146
Series Titles 148
Back Cover 149

2023-05-18