Knot theory / Vassily Manturov.

Format:

Book

Author/Creator:

Manturov, V. O. (Vasiliĭ Olegovich)

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Knot theory.

Physical Description:

400 pages : illustrations ; 25 cm

Place of Publication:

Boca Raton, Fla. : Chapman & Hall/CRC, [2004]

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, Knot Theory describes, with full proofs, the main concepts and most recent advances in the field. The author presents both the "old" theory of knots and the latest investigations by luminaries such as Conway, Matveev, Jones, Kauffman, Vassiliev, Konsevich, Bar-Natan, and Birman. He also explores the most significant results from braid theory, describes braid groups in different spaces, and describes the construction of the Jones two-variable polynomial. One section is devoted to the theory of coding knots by d-diagrams and presents a new method for encoding topological objects by words in a finite alphabet. Another section delves into virtual knot theory and includes 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 DGA construction.

Contents:

I Knots, links, and invariant polynomials 1

2 Reidemeister moves. Knot arithmetics 11

2.1 Polygonal links and Reidemeister moves 11

2.2 Knot arithmetics and Seifert surfaces 15

3 Links in 2-surfaces in R[superscript 3]. Simplest link invariants 25

3.1 Knots in Surfaces. The classification of torus Knots 25

3.2 The linking coefficient 29

3.3 The Arf invariant 31

3.4 The colouring invariant 33

4 Fundamental group. The knot group 37

4.1 Digression. Examples of unknotting 37

4.2 Fundamental group. Basic definitions and examples 40

4.3 Calculating knot groups 45

5 The knot quandle and the Conway algebra 49

5.2 Geometric and algebraic definitions of the quandle 52

5.2.1 Geometric description of the quandle 52

5.2.2 Algebraic description of the quandle 53

5.3 Completeness of the quandle 54

5.4 Special realisations of the quandle: colouring invariant, fundamental group, Alexander polynomial 57

5.5 The Conway algebra and polynomial invariants 57

5.6 Realisations of the Conway algebra. The Conway-Alexander, Jones, HOMFLY and Kauffman polynomials 65

5.7 More on Alexander's polynomial. Matrix representation 66

6 Kauffman's approach to Jones polynomial 69

6.1 State models in physics and Kauffman's bracket 69

6.2 Kauffman's form of Jones polynomial and skein relations 72

6.3 Kauffman's two-variable polynomial 74

7 Properties of Jones polynomials. Khovanov's complex 75

7.1 Simplest properties 75

7.2 Tait's first conjecture and Kauffman-Murasugi's theorem 78

7.3 Menasco-Thistletwaite theorem and the classification of alternating links 79

7.4 The third Tait conjecture 80

7.5 A knot table 80

7.6 Khovanov's categorification of the Jones polynomial 80

7.6.1 The two phenomenological conjectures 88

II Theory of braids 91

8 Braids, links and representations of braid groups 93

8.1 Four definitions of the braid group 93

8.1.1 Geometrical definition 93

8.1.2 Topological definition 94

8.1.3 Algebro-geometrical definition 95

8.1.4 Algebraic definition 95

8.1.5 Equivalence of the four definitions 95

8.1.6 The stable braid group 98

8.1.7 Pure braids 98

8.2 Links as braid closures 103

8.3 Braids and the Jones polynomial 104

8.4 Representations of the braid groups 110

8.4.1 The Burau representation 110

8.4.2 A counterexample 114

8.5 The Krammer-Bigelow representation 115

8.5.1 Krammer's explicit formulae 116

8.5.2 Bigelow's construction and main ideas of the proof 116

9 Braids and links. Braid construction algorithms 121

9.1 Alexander's theorem 121

9.2 Vogel's algorithm 123

10 Algorithms of braid recognition 129

10.1 The curve algorithm for braid recognition 129

10.1.2 Construction of the invariant 130

10.1.3 Algebraic description of the invariant 135

10.2 LD-systems and the Dehornoy algorithm 137

10.2.1 Why the Dehornoy algorithm stops 149

10.3 Minimal word problem for Br(3) 150

10.4 Spherical, cylindrical, and other braids 152

10.4.1 Spherical braids 152

10.4.2 Cylindrical braids 155

11 Markov's theorem. The Yang-Baxter equation 161

11.1 Markov's theorem after Morton 161

11.1.1 Formulation. Definitions. Threadings 161

11.1.2 Markov's theorem and threadings 166

11.2 Makanin's generalisations. Unary braids 174

11.3 Yang-Baxter equation, braid groups and link invariants 175

III Vassiliev's invariants 179

12 Definition and Basic notions of Vassiliev invariant theory 181

12.1 Singular knots and the definition of finite-type invariants 181

12.2 Invariants of orders zero and one 183

12.3 Examples of higher-order invariants 183

12.4 Symbols of Vassiliev's invariants coming from the Conway polynomial 184

12.5 Other polynomials and Vassiliev's invariants 186

12.6 An example of an infinite-order invariant 190

13 The chord diagram algebra 193

13.1 Basic structures 193

13.2 Bialgebra structure of algebras A[superscript c] and A[superscript t]. Chord diagrams and Feynman diagrams 197

13.3 Lie algebra representations, chord diagrams, and the four colour theorem 201

13.4 Dimension estimates for A[subscript d]. A table of known dimensions 204

13.4.1 Historical development 204

13.4.2 An upper bound 205

13.4.3 A lower bound 206

13.4.4 A table of dimensions 208

14 The Kontsevich integral and formulae for the Vassiliev invariants 209

14.1 Preliminary Kontsevich integral 210

14.2 Z([infinity]) and the normalisation 214

14.3 Coproduct for Feynman diagrams 215

14.4 Invariance of the Kontsevich integral 217

14.4.1 Integrating holonomies 218

14.5 Vassiliev's module 227

14.6 Goussarov's theorem 228

14.6.1 Gauss diagrams 228

IV Atoms and d-diagrams 231

15 Atoms, height atoms and knots 233

15.1 Atoms and height atoms 233

15.2 Theorem on atoms and knots 235

15.3 Encoding of knots by d-diagrams 235

15.4 d-diagrams and chord diagrams. Embeddability criterion 239

15.5 A new proof of the Kauffman-Murasugi theorem 243

16 The bracket semigroup of knots 245

16.1 Representation of long links by words in a finite alphabet 245

16.2 Representation of links by quasitoric braids 247

16.2.1 Definition of quasitoric braids 248

16.2.2 Pure braids are quasitoric 249

16.2.3 d-diagrams of quasitoric braids 252

V Virtual knots 255

17.1 Combinatorial definition 257

17.2 Projections from handle bodies 259

17.3 Gauss diagram approach 261

17.4 Virtual knots and links and their simplest invariants 262

17.5 Invariants coming from the virtual quandle 262

17.5.1 Fundamental groups 262

17.5.2 Strange properties of virtual knots 263

18 Invariant polynomials of virtual links 265

18.1 The virtual grouppoid (quandle) 266

18.2 The Jones-Kauffman polynomial 273

18.3 Presentations of the quandle 274

18.3.1 The fundamental group 274

18.3.2 The colouring invariant 275

18.4 The V A-polynomial 276

18.4.1 Properties of the V A-polynomial 281

18.5 Multiplicative approach 283

18.6 The two-variable polynomial 283

18.7 The multivariable polynomial 290

19 Generalised Jones-Kauffman polynomial 293

19.1 Introduction. Basic definitions 293

19.3 Atoms and virtual knots.

Minimality problems 299

20 Long virtual knots and their invariants 303

20.2 The long quandle 304

20.3 Colouring invariant 306

20.4 The [characters not reproducible]-rational function 307

20.5 Virtual knots versus long virtual knots 308

21 Virtual braids 311

21.1 Definitions of virtual braids 311

21.2 Burau representation and its generalisations 312

21.3 Invariants of virtual braids 313

21.3.2 The construction of the main invariant 314

21.3.3 First fruits 316

21.3.4 How strong is the invariant f? 319

21.4 Virtual links as closures of virtual braids 323

21.5 An analogue of Markov's theorem 323

VI Other theories 325

22 3-manifolds and knots in 3-manifolds 327

22.1 Knots in RP[superscript 3] 327

22.2 An introduction to the Kirby theory 330

22.2.1 The Heegaard theorem 330

22.2.2 Constructing manifolds by framed links 332

22.2.3 How to draw bands 333

22.2.4 The Kirby moves 333

22.3 The Witten invariants 335

22.3.1 The Temperley-Lieb algebra 336

22.3.2 The Jones-Wenzl idempotent 339

22.3.3 The main construction 341

22.4 Invariants of links in three-manifolds 344

22.5 Virtual 3-manifolds and their invariants 345

23 Legendrian knots and their invariants 347

23.1 Legendrian manifolds and Legendrian curves 347

23.1.1 Contact structures 347

23.1.2 Planar projections of Legendrian links 348

23.2 Definition, basic notions, and theorems 350

23.3 Fuchs-Tabachnikov moves 352

23.4 Maslov and Bennequin numbers 353

23.5 Finite-type invariants of Legendrian knots 354

23.6 The differential graded algebra (DGA) of a Legendrian knot 355

23.7 Chekanov-Pushkar' invariants 356

A Independence of Reidemeister moves 359

B Vassiliev's invariants for virtual links 363

B.1 The Goussarov-Viro-Polyak approach 363

B.2 The Kauffman approach 364

C Energy of a knot 367

D Unsolved problems in knot theory 371

E A knot table 375.

Notes:

Includes bibliographical references (pages 383-397) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0415310016

OCLC:

53814478