Knot Theory: Second Edition

"Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. It suffices to mention the great progress in knot homology theory (Khovanov homology and Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give rise to strong invariants of knots and 3-manifolds, in particular, many new unknot detectors. New to the Second Edition; discusses Heegaard-Floer homology theory and A-polynomial of classical links. Updates throughout the book."--Provided by publisher.

9.1.7.1 Pure braid groups and mapping classes.

Cover; Half Title; Title; Copyrights; Dedication; Preface; Preface to the second edition; Contents; Part I. Knots, links, and invariant polynomials; Chapter 1. Introduction; 1.1 Basic definitions; Chapter 2. Reidemeister moves. Knot arithmetics; 2.1 Polygonal links and Reidemeister moves; 2.2 Independence of Reidemeister moves; 2.3 Knot arithmetics and Seifert surfaces; Chapter 3. Links in 2-surfaces in R3. Simplest link invariants; 3.1 Knots in 2-surfaces. The classification of torus knots; 3.2 The linking coefficient; 3.3 The Arf invariant; 3.4 The colouring invariant.

Chapter 4. Fundamental group. The knot group4.1 Digression. Examples of unknotting; 4.2 Fundamental group. Basic definitions and examples; 4.3 Calculating knot groups; Chapter 5. The knot quandle and the Conway algebra; 5.1 Introduction; 5.2 Geometric and algebraic definitions of the knot quandle; 5.2.1 Geometric description of the quandle; 5.2.2 Algebraic description of the quandle; 5.3 Completeness of the quandle; 5.4 Special realisations of the quandle: colouring invariant, fundamental group, Alexander polynomial; 5.5 The Conway algebra and polynomial invariants.

5.6 Realisations of the Conway algebra. The Co…