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 polynomials | p. 1 |
Introduction | p. 3 |
Basic definitions | p. 4 |
Reidemeister moves. Knot arithmetics | p. 11 |
Polygonal links and Reidemeister moves | p. 11 |
Knot arithmetics and Seifert surfaces | p. 15 |
Links in 2-surfaces in R[superscript 3]. Simplest link invariants | p. 25 |
Knots in Surfaces. The classification of torus Knots | p. 25 |
The linking coefficient | p. 29 |
The Arf invariant | p. 31 |
The colouring invariant | p. 33 |
Fundamental group. The knot group | p. 37 |
Digression. Examples of unknotting | p. 37 |
Fundamental group. Basic definitions and examples | p. 40 |
Calculating knot groups | p. 45 |
The knot quandle and the Conway algebra | p. 49 |
Introduction | p. 49 |
Geometric and algebraic definitions of the quandle | p. 52 |
Geometric description of the quandle | p. 52 |
Algebraic description of the quandle | p. 53 |
Completeness of the quandle | p. 54 |
Special realisations of the quandle: colouring invariant, fundamental group, Alexander polynomial | p. 57 |
The Conway algebra and polynomial invariants | p. 57 |
Realisations of the Conway algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman polynomials | p. 65 |
More on Alexander's polynomial. Matrix representation | p. 66 |
Kauffman's approach to Jones polynomial | p. 69 |
State models in physics and Kauffman's bracket | p. 69 |
Kauffman's form of Jones polynomial and skein relations | p. 72 |
Kauffman's two-variable polynomial | p. 74 |
Properties of Jones polynomials. Khovanov's complex | p. 75 |
Simplest properties | p. 75 |
Tait's first conjecture and Kauffman-Murasugi's theorem | p. 78 |
Menasco-Thistletwaite theorem and the classification of alternating links | p. 79 |
The third Tait conjecture | p. 80 |
A knot table | p. 80 |
Khovanov's categorification of the Jones polynomial | p. 80 |
The two phenomenological conjectures | p. 88 |
Theory of braids | p. 91 |
Braids, links and representations of braid groups | p. 93 |
Four definitions of the braid group | p. 93 |
Geometrical definition | p. 93 |
Topological definition | p. 94 |
Algebro-geometrical definition | p. 95 |
Algebraic definition | p. 95 |
Equivalence of the four definitions | p. 95 |
The stable braid group | p. 98 |
Pure braids | p. 98 |
Links as braid closures | p. 103 |
Braids and the Jones polynomial | p. 104 |
Representations of the braid groups | p. 110 |
The Burau representation | p. 110 |
A counterexample | p. 114 |
The Krammer-Bigelow representation | p. 115 |
Krammer's explicit formulae | p. 116 |
Bigelow's construction and main ideas of the proof | p. 116 |
Braids and links. Braid construction algorithms | p. 121 |
Alexander's theorem | p. 121 |
Vogel's algorithm | p. 123 |
Algorithms of braid recognition | p. 129 |
The curve algorithm for braid recognition | p. 129 |
Introduction | p. 129 |
Construction of the invariant | p. 130 |
Algebraic description of the invariant | p. 135 |
LD-systems and the Dehornoy algorithm | p. 137 |
Why the Dehornoy algorithm stops | p. 149 |
Minimal word problem for Br(3) | p. 150 |
Spherical, cylindrical, and other braids | p. 152 |
Spherical braids | p. 152 |
Cylindrical braids | p. 155 |
Markov's theorem. The Yang-Baxter equation | p. 161 |
Markov's theorem after Morton | p. 161 |
Formulation. Definitions. Threadings | p. 161 |
Markov's theorem and threadings | p. 166 |
Makanin's generalisations. Unary braids | p. 174 |
Yang-Baxter equation, braid groups and link invariants | p. 175 |
Vassiliev's invariants | p. 179 |
Definition and Basic notions of Vassiliev invariant theory | p. 181 |
Singular knots and the definition of finite-type invariants | p. 181 |
Invariants of orders zero and one | p. 183 |
Examples of higher-order invariants | p. 183 |
Symbols of Vassiliev's invariants coming from the Conway polynomial | p. 184 |
Other polynomials and Vassiliev's invariants | p. 186 |
An example of an infinite-order invariant | p. 190 |
The chord diagram algebra | p. 193 |
Basic structures | p. 193 |
Bialgebra structure of algebras A[superscript c] and A[superscript t]. Chord diagrams and Feynman diagrams | p. 197 |
Lie algebra representations, chord diagrams, and the four colour theorem | p. 201 |
Dimension estimates for A[subscript d]. A table of known dimensions | p. 204 |
Historical development | p. 204 |
An upper bound | p. 205 |
A lower bound | p. 206 |
A table of dimensions | p. 208 |
The Kontsevich integral and formulae for the Vassiliev invariants | p. 209 |
Preliminary Kontsevich integral | p. 210 |
Z([infinity]) and the normalisation | p. 214 |
Coproduct for Feynman diagrams | p. 215 |
Invariance of the Kontsevich integral | p. 217 |
Integrating holonomies | p. 218 |
Vassiliev's module | p. 227 |
Goussarov's theorem | p. 228 |
Gauss diagrams | p. 228 |
Atoms and d-diagrams | p. 231 |
Atoms, height atoms and knots | p. 233 |
Atoms and height atoms | p. 233 |
Theorem on atoms and knots | p. 235 |
Encoding of knots by d-diagrams | p. 235 |
d-diagrams and chord diagrams. Embeddability criterion | p. 239 |
A new proof of the Kauffman-Murasugi theorem | p. 243 |
The bracket semigroup of knots | p. 245 |
Representation of long links by words in a finite alphabet | p. 245 |
Representation of links by quasitoric braids | p. 247 |
Definition of quasitoric braids | p. 248 |
Pure braids are quasitoric | p. 249 |
d-diagrams of quasitoric braids | p. 252 |
Virtual knots | p. 255 |
Basic definitions and motivation | p. 257 |
Combinatorial definition | p. 257 |
Projections from handle bodies | p. 259 |
Gauss diagram approach | p. 261 |
Virtual knots and links and their simplest invariants | p. 262 |
Invariants coming from the virtual quandle | p. 262 |
Fundamental groups | p. 262 |
Strange properties of virtual knots | p. 263 |
Invariant polynomials of virtual links | p. 265 |
The virtual grouppoid (quandle) | p. 266 |
The Jones-Kauffman polynomial | p. 273 |
Presentations of the quandle | p. 274 |
The fundamental group | p. 274 |
The colouring invariant | p. 275 |
The V A-polynomial | p. 276 |
Properties of the V A-polynomial | p. 281 |
Multiplicative approach | p. 283 |
Introduction | p. 283 |
The two-variable polynomial | p. 283 |
The multivariable polynomial | p. 290 |
Generalised Jones-Kauffman polynomial | p. 293 |
Introduction. Basic definitions | p. 293 |
An example | p. 299 |
Atoms and virtual knots. Minimality problems | p. 299 |
Long virtual knots and their invariants | p. 303 |
Introduction | p. 303 |
The long quandle | p. 304 |
Colouring invariant | p. 306 |
The [characters not reproducible]-rational function | p. 307 |
Virtual knots versus long virtual knots | p. 308 |
Virtual braids | p. 311 |
Definitions of virtual braids | p. 311 |
Burau representation and its generalisations | p. 312 |
Invariants of virtual braids | p. 313 |
Introduction | p. 313 |
The construction of the main invariant | p. 314 |
First fruits | p. 316 |
How strong is the invariant f? | p. 319 |
Virtual links as closures of virtual braids | p. 323 |
An analogue of Markov's theorem | p. 323 |
Other theories | p. 325 |
3-manifolds and knots in 3-manifolds | p. 327 |
Knots in RP[superscript 3] | p. 327 |
An introduction to the Kirby theory | p. 330 |
The Heegaard theorem | p. 330 |
Constructing manifolds by framed links | p. 332 |
How to draw bands | p. 333 |
The Kirby moves | p. 333 |
The Witten invariants | p. 335 |
The Temperley-Lieb algebra | p. 336 |
The Jones-Wenzl idempotent | p. 339 |
The main construction | p. 341 |
Invariants of links in three-manifolds | p. 344 |
Virtual 3-manifolds and their invariants | p. 345 |
Legendrian knots and their invariants | p. 347 |
Legendrian manifolds and Legendrian curves | p. 347 |
Contact structures | p. 347 |
Planar projections of Legendrian links | p. 348 |
Definition, basic notions, and theorems | p. 350 |
Fuchs-Tabachnikov moves | p. 352 |
Maslov and Bennequin numbers | p. 353 |
Finite-type invariants of Legendrian knots | p. 354 |
The differential graded algebra (DGA) of a Legendrian knot | p. 355 |
Chekanov-Pushkar' invariants | p. 356 |
Basic examples | p. 358 |
Independence of Reidemeister moves | p. 359 |
Vassiliev's invariants for virtual links | p. 363 |
The Goussarov-Viro-Polyak approach | p. 363 |
The Kauffman approach | p. 364 |
Some observations | p. 365 |
Energy of a knot | p. 367 |
Unsolved problems in knot theory | p. 371 |
A knot table | p. 375 |
Bibliography | p. 383 |
Index | p. 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