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Knot Theory - Vassily Olegovich Manturov


Published: 24th February 2004
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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.

"[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

Knots, links, and invariant polynomialsp. 1
Introductionp. 3
Basic definitionsp. 4
Reidemeister moves. Knot arithmeticsp. 11
Polygonal links and Reidemeister movesp. 11
Knot arithmetics and Seifert surfacesp. 15
Links in 2-surfaces in R[superscript 3]. Simplest link invariantsp. 25
Knots in Surfaces. The classification of torus Knotsp. 25
The linking coefficientp. 29
The Arf invariantp. 31
The colouring invariantp. 33
Fundamental group. The knot groupp. 37
Digression. Examples of unknottingp. 37
Fundamental group. Basic definitions and examplesp. 40
Calculating knot groupsp. 45
The knot quandle and the Conway algebrap. 49
Introductionp. 49
Geometric and algebraic definitions of the quandlep. 52
Geometric description of the quandlep. 52
Algebraic description of the quandlep. 53
Completeness of the quandlep. 54
Special realisations of the quandle: colouring invariant, fundamental group, Alexander polynomialp. 57
The Conway algebra and polynomial invariantsp. 57
Realisations of the Conway algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman polynomialsp. 65
More on Alexander's polynomial. Matrix representationp. 66
Kauffman's approach to Jones polynomialp. 69
State models in physics and Kauffman's bracketp. 69
Kauffman's form of Jones polynomial and skein relationsp. 72
Kauffman's two-variable polynomialp. 74
Properties of Jones polynomials. Khovanov's complexp. 75
Simplest propertiesp. 75
Tait's first conjecture and Kauffman-Murasugi's theoremp. 78
Menasco-Thistletwaite theorem and the classification of alternating linksp. 79
The third Tait conjecturep. 80
A knot tablep. 80
Khovanov's categorification of the Jones polynomialp. 80
The two phenomenological conjecturesp. 88
Theory of braidsp. 91
Braids, links and representations of braid groupsp. 93
Four definitions of the braid groupp. 93
Geometrical definitionp. 93
Topological definitionp. 94
Algebro-geometrical definitionp. 95
Algebraic definitionp. 95
Equivalence of the four definitionsp. 95
The stable braid groupp. 98
Pure braidsp. 98
Links as braid closuresp. 103
Braids and the Jones polynomialp. 104
Representations of the braid groupsp. 110
The Burau representationp. 110
A counterexamplep. 114
The Krammer-Bigelow representationp. 115
Krammer's explicit formulaep. 116
Bigelow's construction and main ideas of the proofp. 116
Braids and links. Braid construction algorithmsp. 121
Alexander's theoremp. 121
Vogel's algorithmp. 123
Algorithms of braid recognitionp. 129
The curve algorithm for braid recognitionp. 129
Introductionp. 129
Construction of the invariantp. 130
Algebraic description of the invariantp. 135
LD-systems and the Dehornoy algorithmp. 137
Why the Dehornoy algorithm stopsp. 149
Minimal word problem for Br(3)p. 150
Spherical, cylindrical, and other braidsp. 152
Spherical braidsp. 152
Cylindrical braidsp. 155
Markov's theorem. The Yang-Baxter equationp. 161
Markov's theorem after Mortonp. 161
Formulation. Definitions. Threadingsp. 161
Markov's theorem and threadingsp. 166
Makanin's generalisations. Unary braidsp. 174
Yang-Baxter equation, braid groups and link invariantsp. 175
Vassiliev's invariantsp. 179
Definition and Basic notions of Vassiliev invariant theoryp. 181
Singular knots and the definition of finite-type invariantsp. 181
Invariants of orders zero and onep. 183
Examples of higher-order invariantsp. 183
Symbols of Vassiliev's invariants coming from the Conway polynomialp. 184
Other polynomials and Vassiliev's invariantsp. 186
An example of an infinite-order invariantp. 190
The chord diagram algebrap. 193
Basic structuresp. 193
Bialgebra structure of algebras A[superscript c] and A[superscript t]. Chord diagrams and Feynman diagramsp. 197
Lie algebra representations, chord diagrams, and the four colour theoremp. 201
Dimension estimates for A[subscript d]. A table of known dimensionsp. 204
Historical developmentp. 204
An upper boundp. 205
A lower boundp. 206
A table of dimensionsp. 208
The Kontsevich integral and formulae for the Vassiliev invariantsp. 209
Preliminary Kontsevich integralp. 210
Z([infinity]) and the normalisationp. 214
Coproduct for Feynman diagramsp. 215
Invariance of the Kontsevich integralp. 217
Integrating holonomiesp. 218
Vassiliev's modulep. 227
Goussarov's theoremp. 228
Gauss diagramsp. 228
Atoms and d-diagramsp. 231
Atoms, height atoms and knotsp. 233
Atoms and height atomsp. 233
Theorem on atoms and knotsp. 235
Encoding of knots by d-diagramsp. 235
d-diagrams and chord diagrams. Embeddability criterionp. 239
A new proof of the Kauffman-Murasugi theoremp. 243
The bracket semigroup of knotsp. 245
Representation of long links by words in a finite alphabetp. 245
Representation of links by quasitoric braidsp. 247
Definition of quasitoric braidsp. 248
Pure braids are quasitoricp. 249
d-diagrams of quasitoric braidsp. 252
Virtual knotsp. 255
Basic definitions and motivationp. 257
Combinatorial definitionp. 257
Projections from handle bodiesp. 259
Gauss diagram approachp. 261
Virtual knots and links and their simplest invariantsp. 262
Invariants coming from the virtual quandlep. 262
Fundamental groupsp. 262
Strange properties of virtual knotsp. 263
Invariant polynomials of virtual linksp. 265
The virtual grouppoid (quandle)p. 266
The Jones-Kauffman polynomialp. 273
Presentations of the quandlep. 274
The fundamental groupp. 274
The colouring invariantp. 275
The V A-polynomialp. 276
Properties of the V A-polynomialp. 281
Multiplicative approachp. 283
Introductionp. 283
The two-variable polynomialp. 283
The multivariable polynomialp. 290
Generalised Jones-Kauffman polynomialp. 293
Introduction. Basic definitionsp. 293
An examplep. 299
Atoms and virtual knots. Minimality problemsp. 299
Long virtual knots and their invariantsp. 303
Introductionp. 303
The long quandlep. 304
Colouring invariantp. 306
The [characters not reproducible]-rational functionp. 307
Virtual knots versus long virtual knotsp. 308
Virtual braidsp. 311
Definitions of virtual braidsp. 311
Burau representation and its generalisationsp. 312
Invariants of virtual braidsp. 313
Introductionp. 313
The construction of the main invariantp. 314
First fruitsp. 316
How strong is the invariant f?p. 319
Virtual links as closures of virtual braidsp. 323
An analogue of Markov's theoremp. 323
Other theoriesp. 325
3-manifolds and knots in 3-manifoldsp. 327
Knots in RP[superscript 3]p. 327
An introduction to the Kirby theoryp. 330
The Heegaard theoremp. 330
Constructing manifolds by framed linksp. 332
How to draw bandsp. 333
The Kirby movesp. 333
The Witten invariantsp. 335
The Temperley-Lieb algebrap. 336
The Jones-Wenzl idempotentp. 339
The main constructionp. 341
Invariants of links in three-manifoldsp. 344
Virtual 3-manifolds and their invariantsp. 345
Legendrian knots and their invariantsp. 347
Legendrian manifolds and Legendrian curvesp. 347
Contact structuresp. 347
Planar projections of Legendrian linksp. 348
Definition, basic notions, and theoremsp. 350
Fuchs-Tabachnikov movesp. 352
Maslov and Bennequin numbersp. 353
Finite-type invariants of Legendrian knotsp. 354
The differential graded algebra (DGA) of a Legendrian knotp. 355
Chekanov-Pushkar' invariantsp. 356
Basic examplesp. 358
Independence of Reidemeister movesp. 359
Vassiliev's invariants for virtual linksp. 363
The Goussarov-Viro-Polyak approachp. 363
The Kauffman approachp. 364
Some observationsp. 365
Energy of a knotp. 367
Unsolved problems in knot theoryp. 371
A knot tablep. 375
Bibliographyp. 383
Indexp. 398
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780415310017
ISBN-10: 0415310016
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 416
Published: 24th February 2004
Publisher: Taylor & Francis Ltd
Country of Publication: GB
Dimensions (cm): 23.5 x 16.51  x 3.18
Weight (kg): 0.73
Edition Number: 1