Knot Theory: Second Edition, 2nd Edition (Hardback) book cover

Knot Theory

Second Edition, 2nd Edition

By Vassily Olegovich Manturov

CRC Press

560 pages | 289 B/W Illus.

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Hardback: 9781138561243
pub: 2018-03-29
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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.

Reviews

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

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 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

About the Author

Vassily Olegovich Manturov is professor of Geometry and Topology at Bauman Moscow State Technical University.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT012000
MATHEMATICS / Geometry / General
SCI040000
SCIENCE / Mathematical Physics
SCI055000
SCIENCE / Physics