Knot Theory

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. Praise for the first edition This book is highly recommended for all students and researchers in knot theory, and to those in the sciences and mathematics who would like to get a flavor of this very active field." -Professor Louis H. Kauffman, Department of Mathematics, Statistics and Com-puter Science, University of Illinois at Chicago

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 Conjecture II Theory of braids 9 Braids, links and representations 10 Braids and links 11 Algorithms of braid recognition 12 Markov's theorem. YBE III Vassiliev's invariants. Atoms and d-diagrams 13 Definition and Basic notions 14 The chord diagram algebra 15 Kontsevich's integral 16 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 knot B TheA-Polynomial C Garside's Normal Form D Unsolved problems in knot theory