Knot Theory: Second Edition

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 POLYNOMIALSINTRODUCTIONBasic DefinitionsREIDEMEISTER MOVES. KNOT ARITHMETICSPolygonal Links and Reidemeister MovesKnot Arithmetics and Seifert SurfacesLINKS IN 2 SURFACES IN R^3. SIMPLEST LINK INVARIANTSKnots in Surfaces. The Classiffcation of Torus KnotsThe Linking CoefficientThe Arf InvariantThe Colouring InvariantFUNDAMENTAL GROUP. THE KNOT GROUPDigression. Examples of UnknottingFundamental Group. Basic Definitions and ExamplesCalculating Knot GroupsTHE KNOT QUANDLE AND THE CONWAY ALGEBRAIntroductionGeometric and Algebraic Definitions of the QuandleCompleteness of the QuandleSpecial Realisations of the Quandle: Colouring Invariant, Fundamental Group, Alexander PolynomialThe Conway Algebra and Polynomial InvariantsRealisations of the Conway Algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman PolynomialsMore on Alexander's polynomial. Matrix representationKAUFFMAN'S APPROACH TO JONES POLYNOMIALState models in Physics and Kauffman's BracketKauffman's Form of Jones Polynomial and Skein RelationsKauffman's Two-Variable PolynomialPROPERTIES OF JONES POLYNOMIALS. KHOVANOV'S COMPLEXSimplest PropertiesTait's First Conjecture and Kauffman-Murasugi's TheoremMenasco-Thistletwaite Theorem and the Classification of Alternating LinksThe Third Tait ConjectureA Knot TableKhovanov's Categorification of the Jones PolynomialThe Two Phenomenological ConjecturesII. THEORY OF BRAIDSBraids, Links and Representations of Braid GroupsFour Definitions of the Braid GroupLinks as Braid ClosuresBraids and the Jones PolynomialRepresentations of the Braid GroupsThe Krammer-Bigelow RepresentationBRAIDS AND LINKS. BRAID CONSTRUCTION ALGORITHMSAlexander's TheoremVogel's AlgorithmALGORITHMS OF BRAID RECOGNITIONThe Curve Algorithm for Braid RecognitionLD-Systems and the Dehornoy AlgorithmMinimal Word Problem for Br(3)Spherical, Cylindrical, and other BraidsMARKOV'S THEOREM. THE YANG-BAXTER EQUATIONMarkov's Theorem after MORTONMakanin's Generalisations. Unary BraidsYang-Baxter Equation, Braid Groups and Link InvariantsIII. VASSILIEV'S INVARIANTS Definition and Basic Notions of Vassiliev Invariant TheorySingular Knots and the Definition of Finite-Type InvariantsInvariants of Orders Zero and OneExamples of Higher-Order InvariantsSymbols of Vassiliev's Invariants Coming from the Conway PolynomialOther Polynomials and Vassiliev's InvariantsAn Example of an Infinite-Order InvariantTHE CHORD DIAGRAM ALGEBRABasic StructuresBialgebra Structure of Algebras A^c and A^t. Chord Diagrams and Feynman diagramsLie Algebra Representations, Chord Diagrams, and the Four Colour TheoremDimension estimates for Ad. A Table of Known DimensionsTHE KONTSEVICH INTEGRAL AND FORMULAE FOR THE VASSILIEV INVARIANTS209Preliminary Kontsevich IntegralZ(8) and the NormalisationCoproduct for Feynman DiagramsInvariance of the Kontsevich IntegralVassiliev's ModuleGoussarov's TheoremIV. ATOMS AND d-DIAGRAMSATOMS, HEIGHT ATOMS AND KNOTSAtoms and Height AtomsTheorem on Atoms and KnotsEncoding of Knots by d-diagramsd-Diagrams and Chord Diagrams. Embeddability CriterionA New Proof of the Kauffman-Murasugi TheoremTHE BRACKET SEMIGROUP OF KNOTSRepresentation of Long Links by Words in a Finite AlphabetRepresentation of Links by Quasitoric BraidsV. VIRTUAL KNOTSBASIC DEFINITIONS AND MOTIVATIONCombinatorial DefinitionProjections from Handle BodiesGauss Diagram ApproachVirtual Knots and Links and their Simplest InvariantsInvariants Coming from the Virtual QuandleINVARIANT POLYNOMIALS OF VIRTUAL LINKSThe Virtual Grouppoid (Quandle)The Jones-Kauffman PolynomialPresentations of the QuandleThe V A-PolynomialProperties of the V A-PolynomialMultiplicative ApproachThe Two-Variable PolynomialThe Multivariable PolynomialGENERALISED JONES-KAUFFMAN POLYNOMIALIntroduction. Basic DefinitionsAn ExampleAtoms and Virtual Knots. Minimality ProblemsLONG VIRTUAL KNOTS AND THEIR INVARIANTSIntroductionThe Long QuandleColouring InvariantThe V-Rational FunctionVirtual Knots versus Long Virtual KnotsVIRTUAL BRAIDSDefinitions of Virtual BraidsBurau Representation and its GeneralisationsInvariants of Virtual BraidsVirtual Links as Closures of Virtual BraidsAn Analogue of Markov's TheoremVI. OTHER THEORIES3-MANIFOLDS AND KNOTS IN 3-MANIFOLDSKnots in RP^3An Introduction to the Kirby TheoryThe Witten InvariantsInvariants of Links in Three-ManifoldsVirtual 3-Manifolds and their InvariantsLEGENDRIAN KNOTS AND THEIR INVARIANTSLegendrian Manifolds and Legendrian CurvesDefinition, Basic Notions, and TheoremsFuchs-Tabachnikov MovesMaslov and Bennequin NumbersFinite-type Invariants of Legendrian KnotsThe Differential Graded Algebra (DGA) of a Legendrian KnotChekanov-Pushkar' InvariantsBasic ExamplesAPPENDICESIndependence of Reidemeister MovesVassiliev's Invariants for Virtual LinksEnergy of a KnotUnsolved Problems in Knot TheoryA Knot TableBIBLIOGRAPHYINDEX

Review :
"[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