Knot Theory

Description

Title page ACKNOWLEDGEMENTS PREFACE Chapter 1 A CENTURY OF KNOT THEORY Chapter 2 WHAT IS A KNOT? Section 1 Wild Knots and Unknottings Section 2 The Definition of a Knot Section 9 Equivalence of Knots, Deformations Section 4 Diagrams and Projections Section 5 Orientations Chapter 3 COMBINATORIAL TECHNIQUES Section 1 Reidemeister Moves Section 2 Colorings Section 3 A Generalization of Colorability, mod p Labelings Section 4. Matrices, Labelings, and Determinants Section 5 The Alexander Polynomial Chapter 4 GEOMETRIC TECHNIQUES Section 1 Surfaces and Homeomorphisms Section 2 The Classification of Surfaces Section 3 Seifert Surfaces and the Genus of a Knot Section 4. Surgery on Surfaces Section 5 Connected Sums of Knots and Prime Decompositions Chapter 5 ALGEBRAIC TECHNIQUES Section 1 Symmetric Groups Section 2 Knots and Groups Section 3 Conjugation and the Labeling Theorem Section 4. Equations in Groups and the Group of a Knot Section 5 The Fundamental Group Chapter 6 GEOMETRY, ALGEBRA, AND THE ALEXANDER POLYNOMIAL Section 1 The Seifert Matrix Section 2 Seifert Matrices and the Alexander Polynomial Section 3 The Signature of a Knot, and other S-Equivalence Invariants Section 4. Knot Groups and the Alexander Polynomial Chapter 7 NUMERICAL INVARIANTS Section 1 Summary of Numerical Invariants Section 2 New Invariants Section 3 Braids and Bridges Section 4. Relations Between the Numerical Invariants Section 5 Independence of Numerical Invariants Chapter 8 SYMMETRIES OF KNOTS Section 1 Amphicheiral and Reversible Knots Section 2 Periodic Knots Section 3 The Murasugi Conditions Section 4 Periodic Seifert Surfaces and Edmonds' Theorem Section 5 Applications of the Murasugi and Edmonds Conditions Chapter 9 HIGH-DIMENSIONAL KNOT THEORY Section 1 Defining High-dimensional Knots Section 2 Three Dimensions from a 2-dimensional Perspective Section 3 Three-dimensional Cross-sections of a 4-dimensional Knot Section 4. Slice Knots Section 5 The Knot Concordance Group Chapter 10 NEW COMBINATORIAL TECHNIQUES Section 1 The Conway Polynomial of a Knot Section 2 New Polynomial Invariants Section 3 Kauffman's Bracket Polynomial Appendix 1 KNOT TABLE Appendix 2 ALEXANDER POLYNOMIALS REFERENCES INDEX