Knot Theory

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