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.
Author(s): Alekseĭ Bronislavovich Sosinskiĭ
Edition: 1
Publisher: American Mathematical Society
Year: 2023
Language: English
Commentary: 2020 Mathematics Subject Classification. Primary 55-xx, 51-xx, 20-xx.
Pages: 142
City: Providence, Rhode Island
Tags: Knot Theory; Link Theory; Algebraic Topology; Geometry; Group Theory
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 ?(mskip 2??⋅mskip 2??)
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 ? a Laurent polynomial in ??
6.8. Knot tables revisited
6.9. Exercises
Lecture 7. Braids
7.1. Geometric braids
7.2. The geometric braid group ?_{?}
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 ?_{?}
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 ?=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
