Knot Theory
Features
Summary
Since discovery of the Jones polynomial, knot theory has enjoyed a virtual explosion of important results and now plays a significant role in modern mathematics. In a unique presentation with contents not found in any other monograph, Knot Theory describes, with full proofs, the main concepts and the latest investigations in the field.
The book is divided into six thematic sections. The first part discusses "pre-Vassiliev" knot theory, from knot arithmetics through the Jones polynomial and the famous Kauffman-Murasugi theorem. The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. A section devoted to the Vassiliev knot invariants follows, wherein the author proves that Vassiliev invariants are stronger than all polynomial invariants and introduces Bar-Natan's theory on Lie algebra respresentations and knots.
The fourth part describes a new way, proposed by the author, to encode knots by d-diagrams. This method allows the encoding of topological objects by words in a finite alphabet. Part Five delves into virtual knot theory and virtualizations of knot and link invariants. This section includes the author's own important results regarding new invariants of virtual knots. The book concludes with an introduction to knots in 3-manifolds and Legendrian knots and links, including Chekanov's differential graded algebra (DGA) construction.
Knot Theory 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 professional reference and will serve equally well as a text for a course on knot theory.
Table of Contents
I. KNOTS, LINKS, AND INVARIANT POLYNOMIALS
INTRODUCTION
Basic Definitions
REIDEMEISTER MOVES. KNOT ARITHMETICS
Polygonal Links and Reidemeister Moves
Knot Arithmetics and Seifert Surfaces
LINKS IN 2 SURFACES IN R^3. SIMPLEST LINK INVARIANTS
Knots in Surfaces. The Classiffcation of Torus Knots
The Linking Coefficient
The Arf Invariant
The Colouring Invariant
FUNDAMENTAL GROUP. THE KNOT GROUP
Digression. Examples of Unknotting
Fundamental Group. Basic Definitions and Examples
Calculating Knot Groups
THE KNOT QUANDLE AND THE CONWAY ALGEBRA
Introduction
Geometric and Algebraic Definitions of the Quandle
Completeness of the Quandle
Special Realisations of the Quandle: Colouring Invariant, Fundamental Group, Alexander Polynomial
The Conway Algebra and Polynomial Invariants
Realisations of the Conway Algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman Polynomials
More on Alexander's polynomial. Matrix representation
KAUFFMAN'S APPROACH TO JONES POLYNOMIAL
State models in Physics and Kauffman's Bracket
Kauffman's Form of Jones Polynomial and Skein Relations
Kauffman's Two-Variable Polynomial
PROPERTIES OF JONES POLYNOMIALS. KHOVANOV'S COMPLEX
Simplest Properties
Tait's First Conjecture and Kauffman-Murasugi's Theorem
Menasco-Thistletwaite Theorem and the Classification of Alternating Links
The Third Tait Conjecture
A Knot Table
Khovanov's Categorification of the Jones Polynomial
The Two Phenomenological Conjectures
II. THEORY OF BRAIDS
Braids, Links and Representations of Braid Groups
Four Definitions of the Braid Group
Links as Braid Closures
Braids and the Jones Polynomial
Representations of the Braid Groups
The Krammer-Bigelow Representation
BRAIDS AND LINKS. BRAID CONSTRUCTION ALGORITHMS
Alexander's Theorem
Vogel's Algorithm
ALGORITHMS OF BRAID RECOGNITION
The Curve Algorithm for Braid Recognition
LD-Systems and the Dehornoy Algorithm
Minimal Word Problem for Br(3)
Spherical, Cylindrical, and other Braids
MARKOV'S THEOREM. THE YANG-BAXTER EQUATION
Markov's Theorem after MORTON
Makanin's Generalisations. Unary Braids
Yang-Baxter Equation, Braid Groups and Link Invariants
III. VASSILIEV'S INVARIANTS
Definition and Basic Notions of Vassiliev Invariant Theory
Singular Knots and the Definition of Finite-Type Invariants
Invariants of Orders Zero and One
Examples of Higher-Order Invariants
Symbols of Vassiliev's Invariants Coming from the Conway Polynomial
Other Polynomials and Vassiliev's Invariants
An Example of an Infinite-Order Invariant
THE CHORD DIAGRAM ALGEBRA
Basic Structures
Bialgebra Structure of Algebras A^c and A^t. Chord Diagrams and Feynman diagrams
Lie Algebra Representations, Chord Diagrams, and the Four Colour Theorem
Dimension estimates for Ad. A Table of Known Dimensions
THE KONTSEVICH INTEGRAL AND FORMULAE FOR THE VASSILIEV INVARIANTS209
Preliminary Kontsevich Integral
Z(8) and the Normalisation
Coproduct for Feynman Diagrams
Invariance of the Kontsevich Integral
Vassiliev's Module
Goussarov's Theorem
IV. ATOMS AND d-DIAGRAMS
ATOMS, HEIGHT ATOMS AND KNOTS
Atoms and Height Atoms
Theorem on Atoms and Knots
Encoding of Knots by d-diagrams
d-Diagrams and Chord Diagrams. Embeddability Criterion
A New Proof of the Kauffman-Murasugi Theorem
THE BRACKET SEMIGROUP OF KNOTS
Representation of Long Links by Words in a Finite Alphabet
Representation of Links by Quasitoric Braids
V. VIRTUAL KNOTS
BASIC DEFINITIONS AND MOTIVATION
Combinatorial Definition
Projections from Handle Bodies
Gauss Diagram Approach
Virtual Knots and Links and their Simplest Invariants
Invariants Coming from the Virtual Quandle
INVARIANT POLYNOMIALS OF VIRTUAL LINKS
The Virtual Grouppoid (Quandle)
The Jones-Kauffman Polynomial
Presentations of the Quandle
The V A-Polynomial
Properties of the V A-Polynomial
Multiplicative Approach
The Two-Variable Polynomial
The Multivariable Polynomial
GENERALISED JONES-KAUFFMAN POLYNOMIAL
Introduction. Basic Definitions
An Example
Atoms and Virtual Knots. Minimality Problems
LONG VIRTUAL KNOTS AND THEIR INVARIANTS
Introduction
The Long Quandle
Colouring Invariant
The V-Rational Function
Virtual Knots versus Long Virtual Knots
VIRTUAL BRAIDS
Definitions of Virtual Braids
Burau Representation and its Generalisations
Invariants of Virtual Braids
Virtual Links as Closures of Virtual Braids
An Analogue of Markov's Theorem
VI. OTHER THEORIES
3-MANIFOLDS AND KNOTS IN 3-MANIFOLDS
Knots in RP^3
An Introduction to the Kirby Theory
The Witten Invariants
Invariants of Links in Three-Manifolds
Virtual 3-Manifolds and their Invariants
LEGENDRIAN KNOTS AND THEIR INVARIANTS
Legendrian Manifolds and Legendrian Curves
Definition, Basic Notions, and Theorems
Fuchs-Tabachnikov Moves
Maslov and Bennequin Numbers
Finite-type Invariants of Legendrian Knots
The Differential Graded Algebra (DGA) of a Legendrian Knot
Chekanov-Pushkar' Invariants
Basic Examples
APPENDICES
Independence of Reidemeister Moves
Vassiliev's Invariants for Virtual Links
Energy of a Knot
Unsolved Problems in Knot Theory
A Knot Table
BIBLIOGRAPHY
INDEX
Editorial Reviews
"[This book] can be used as a textbook; it is also intended to serve as a reference on recent developments in knot theory. … [T]he book is rather readable and serves its purpose well."
- Mathematical Reviews, Issue 2005d
"This book is an excellent and up to date introduction to knot theory containing many topics that have not yet appeared in any text book about knots. These topics include the Khovanov generalization of the Jones polynomial, the Krammer-Bigelow faithful representation of the braid group, a systematic treatment of algorithms for braid recognition, the author's theory of atoms and d-diagrams, the theory of virtual knots and the author's theory of long virtual knots, virtual braids and Legendrian knots. Well-known topics are treated as well, with a systematic and well-organized progression of techniques and ideas. 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 Computer Science, University of Illinois at Chicago
