Knots, links and their invariants : an elementary course in contemporary knot theory

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. 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