Description
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 alge-bra. 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 this Edition is a discussion of Heegaard-Floer homology theory and A-polynomial of classical links, as well as updates throughout the text.
Knot Theory, Second Edition is notable not only for its expert presentation of knot theory’s state of the art but also for its accessibility. It is valuable as a profes-sional reference and will serve equally well as a text for a course on knot theory.
Table of Contents
Preface
Preface to the second edition
I Knots, links, and invariant polynomials
1 Introduction
2 Reidemeister moves. Knot arithmetics
3 Torus Knots
4 Fundamental group
5 Quandle and Conway’s algebra
6 Kauffman’s approach to Jones polynomial
7 Jones’ polynomial. Khovanov’s complex
8 Lee-Rasmussen Invariant, Slice Knots, and the Genus ConjectureII Theory of braids
9 Braids, links and representations
10 Braids and links
11 Algorithms of braid recognition
12 Markov’s theorem. YBEIII Vassiliev’s invariants. Atoms and d-diagrams
13 Definition and Basic notions
14 The chord diagram algebra
15 Kontsevich’s integral16 Atoms, height atoms and knots
IV Virtual knots
17 Basic definitions
18 Invariant polynomials of virtual links
19 Generalised Jones–Kauffman polynomial
20 Long Virtual Knots
21 Virtual braids
22 Khovanov Homology of Virtual Knots
V Knots,3-Manifolds, and Legendrian Knots
23 3-Manifolds and knots in 3-manifolds
24 Heegaard-Floer Homology
25 Legendrian knots and their invariants
Appendicies
A Energy of a knotB TheA-Polynomial
C Garside’s Normal Form
D Unsolved problems in knot theory
