Virtual Knots: The State Of The Art

by Vassily Olegovich Manturov, Denis Petrovich Ilyutko

Overview

The book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory.Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory.In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V O Manturov and graph-link theory due to both authors. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams.Graph-links can be treated as “diagramless knot theory”: such “links” have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.

Product Details

ISBN-13:	9789814401128
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/28/2012
Series:	Series On Knots And Everything , #51
Pages:	552
Product dimensions:	6.00(w) x 9.10(h) x 1.30(d)

Table of Contents

Dedication v

Preface Louis H. Kauffman ix

Preface Vassily Olegovich Manturov Denis Petrovich Ilyutko xiii

Acknowledgments xix

1 Basic Definitions and Notions 1

1.1 Classical knots 1

1.2 Virtual knots 8

1.3 Self-linking number 25

2 Virtual Knots and Three-Dimensional Topology 29

2.1 Introduction 29

2.2 The Kuperberg theorem 32

2.3 Genus of a virtual knot 36

2.3.1 Two types of connected sums 41

2.3.2 The proof plan of Theorem 2.5 42

2.3.3 The process of destabilization 44

2.4 Recognition of virtual links 49

3 Quandles (Distributive Groupoids) in Virtual Knot Theory 57

3.1 Introduction 57

3.2 Quandles and their generalizations 60

3.2.1 Geometric description of the quandle 63

3.2.2 Algebraic description of the quandle 64

3.2.3 The virtual quandle 68

3.2.4 The coloring invariant 77

3.2.5 The Alexander virtual module 79

3.3 Long virtual knots 94

3.4 Virtual knots and infinite-dimensional Lie algebras 104

3.4.1 Preliminaries 104

3.4.2 Generalizations 107

3.5 Hierarchy of virtual knots 110

3.5.1 Flat virtual knots 110

3.5.2 Algebraic formalism 112

3.5.3 Examples 117

4 The Jones-Kauffman Polynomial: Atoms 121

4.1 Introduction 121

4.2 Basic definitions 122

4.2.1 Virtualization and mutation 125

4.2.2 Atoms and knots 128

4.2.3 Virtual diagrams and atoms 130

4.2.4 Chord diagrams 132

4.2.5 Passage from atoms to chord diagrams 134

4.2.6 Spanning tree for the Kauffman bracket polynomial 138

4.3 The polynomial Ξ: minimality problems 142

4.3.1 The leading and lowest terms of the Kauffman bracket polynomial 150

4.3.2 The polynomial Ξ 152

4.3.3 Examples of applications of the polynomial Ξ 158

4.3.4 A surface bracket and the invariant Ξ 161

4.4 Rigid virtual knots 164

4.4.1 Kauffman bracket for rigid knots 165

4.4.2 Minimality properties 167

4.5 Minimal diagrams of long virtual knots 168

5 Khovanov Homology 177

5.1 Introduction 177

5.2 Basic constructions: The Jones polynomial J 182

5.3 Khovanov homology with Z₂-coefficients 184

5.4 Khovanov homology of double knots 195

5.5 Khovanov homology and atoms 206

5.6 Khovanov homology and parity 214

5.7 Khovanov homology for virtual links 214

5.7.1 Atoms and twisted virtual knots 214

5.7.2 Khovanov complex for virtual knots 217

5.8 Spanning tree for Khovanov complex 243

5.9 The Khovanov polynomial and Frobenius extensions 245

5.9.1 Frobenius extensions 245

5.9.2 Khovanov construction for Frobenius extensions 246

5.9.3 Geometrical generalizations by means of atoms 248

5.9.4 Algebraic generalizations 249

5.10 Minimal diagrams of links 254

6 Virtual Braids 257

6.1 Introduction 257

6.2 Definitions of virtual braids 257

6.3 Virtual braids and virtual knots 261

6.3.1 Closure of virtual braids 261

6.3.2 Burau representation and its generalizations 266

6.4 The Kauffman bracket polynomial for braids 269

6.5 Invariants of virtual braids 270

6.5.1 The construction of the main invariant 271

6.5.2 Representation of virtual braid group 274

6.5.3 On completeness in the classical case 275

6.5.4 First fruits 275

6.5.5 Completeness for the case of two-strand braids 277

7 Vassiliev's Invariants and Framed Graphs 281

7.1 Introduction 281

7.2 The Vassiliev invariants of classical knots and J-invariants of curves 285

7.3 The Goussarov-Polyak-Viro approach 293

7.4 The Kauffman approach 302

7.4.1 Main definitions 302

7.4.2 Invariants generated by the polynomial Ξ 303

7.5 Vassiliev's invariants coming from the invariant Ξ 305

7.6 Infinity of the number of long virtual knots 307

7.7 Graphs, chord diagrams and the Kauffman polynomial 308

7.8 Euler tours, Gauss circuits and rotating circuits 311

7.8.1 4-Graphs and Euler tours 313

7.8.2 Framed 4-valent graphs and Euler tours 315

7.8.3 The existence of a Gauss circuit 320

7.8.4 The Gauss circuit 321

7.8.5 Adjacency matrices 328

7.9 A proof of Vassiliev's conjecture 337

7.10 Embeddings of framed 4-graphs 345

8 Parity in Knot Theory: Free-Knots: Cobordisms 351

8.1 Introduction 351

8.2 Free knots and parity 354

8.2.1 Free links 355

8.2.2 The parity axiomatics 359

8.2.3 Gaussian parity for free, flat and virtual knots 362

8.2.4 Two-component classical and virtual links 364

8.2.5 Knots in the solid torus, curves on 2-surfaces 364

8.2.6 Parity and homology 365

8.2.7 The universal parity: The classification of parities for free knots 368

8.3 A functorial mapping f 377

8.3.1 Construction 377

8.3.2 The parity from Sec. 8.2.3 378

8.3.3 The parity hierarchy on virtual knots 382

8.3.4 The parity from Sec. 8.2.4 385

8.3.5 The parity from Sec. 8.2.5 385

8.4 Invariants 385

8.4.1 Preliminaries: smoothings and linear spaces 385

8.4.2 The invariants [·], {·} 388

8.4.3 Non-invertibility of free links 397

8.5 Goldman's bracket and Turaev's cobracket 402

8.5.1 The map m: Γ²_S → Γ_S 404

8.5.2 Goldman's Lie algebra 404

8.5.3 The maps Γ_S → Γ²_S,0 and Γ_S → Γ_S ⊗ Γ_S/(triv.) Turaev's cobracket 405

8.6 Applications of Turaev's Delta 406

8.6.1 Non-invertibility of free knots 407

8.6.2 Even and odd analogues of Goldman's bracket and Turaev's cobracket 408

8.7 An analogue of the Kauffman bracket 410

8.8 Virtual crossing numbers for virtual knots 413

8.9 Cobordisms of free knots 419

8.9.1 Introduction 419

8.9.2 Combinatorial cobordism of free knots 421

8.9.3 An invariant of free knots 426

8.9.4 Slice genus and cobordisms of free knots 433

8.9.5 Parity of curves in 2-surfaces 437

8.9.6 Sliceness of free knots 442

8.9.7 Cobordisms of higher genus 448

9 Theory of Graph-Links 451

9.1 Introduction 451

9.2 Graph-links and looped graphs 457

9.2.1 Chord diagrams 457

9.2.2 Reidemeister moves for looped interlacement graphs and graph-links 458

9.2.3 Looped graphs and graph-links 465

9.2.4 Smoothing operations and Turaev's Δ 479

9.3 Parity, minimality and non-trivial examples 482

9.3.1 Definition of parity 482

9.3.2 The universal parity 484

9.3.3 Minimality 487

9.4 A generalization of Kauffman's bracket and other invariants. Minimality theorems 490

Bibliography 499

Index 515