Survey on knot theory

Table of contents......Page 3
Preface......Page 7
A prelude to the study of knot theory......Page 9
Notes on research conventions and notations......Page 18
0.1 Spaces......Page 20
0.2 Manifolds and submanifolds......Page 21
0.3 Knots and links......Page 23
Supplementary notes for Chapter 0......Page 24
1.1 Regular presentations......Page 26
1.2 Braid presentations......Page 31
1.3 Bridge presentations......Page 37
Supplementary notes for Chapter 1......Page 38
2.1 Two-bridge links......Page 39
2.2 Torus links......Page 44
2.3 Pretzel links......Page 45
Supplementary notes for Chapter 2......Page 46
3.1 Compositions of links......Page 48
3.2 Decompositions of links......Page 50
3.3 Definition of a tangle and examples......Page 51
3.4 How to judge the non-splittability of a link......Page 53
3.5 How to judge the primeness of a link......Page 55
3.6 How to judge the hyperbolicity of a link......Page 58
3.7 Non-triviality of a link......Page 59
3.8 Conway mutation......Page 60
Supplementary notes for Chapter 3......Page 62
4.1 Definition and existence of Seifert surfaces......Page 63
4.2 The Murasugi sum......Page 66
4.3 Sutured manifolds......Page 69
Supplementary notes for Chapter 4......Page 75
5.1 The Seifert matrix......Page 77
5.2 S-equivalence......Page 79
5.3 Number-theoretic invariants......Page 81
5.4 The reduced link module......Page 85
5.5 The homology of a branched cyclic covering manifold......Page 86
Supplementary notes for Chapter 5......Page 88
6.1 Link groups and link group systems......Page 89
6.2 Presentations of a link group......Page 94
6.3 Subgroups and quotient groups of a link group......Page 99
Supplementary notes for Chapter 6......Page 102
7.1 The Alexander module......Page 103
7.2 Invariants of a A-module......Page 107
7.3 Graded Alexander polynomials......Page 109
7.4 Torres conditions......Page 113
Supplementary notes for Chapter 7......Page 114
8.1 The Jones polynomial......Page 115
8.2 The skein polynomial......Page 119
8.3 The Q and Kauffman polynomials......Page 122
8.4 Properties of the polynomial invariants......Page 123
8.5 The skein polynomial via a state model......Page 127
Supplementary notes for Chapter 8......Page 128
9.1 Preliminaries from representation theory......Page 129
9.2 Link invariants of trace type......Page 131
9.3 The skein polynomial as a link invariant of trace type......Page 132
9.4 The Temperley-Lieb algebra......Page 133
Supplementary notes for Chapter 9......Page 135
10.1 Periodic knots......Page 136
10.2 Freely periodic knots......Page 140
10.3 Invertible knots......Page 142
10.4 Amphicheiral knots......Page 143
10.5 Symmetries of a hyperbolic knot......Page 144
10.6 The symmetry group......Page 146
10.7 Canonical decompositions and symmetry......Page 149
Supplementary notes for Chapter 10......Page 155
11.1 Unknotting operations......Page 156
11.2 Properties of X-Gordian distance......Page 161
11.3 Properties of \\Delta-Gordian distance......Page 163
11.4 Properties of #-Gordian distance......Page 164
11.5 Estimation of the X-unknotting number......Page 165
11.6 Local transformations of links......Page 166
Supplementary notes for Chapter 11......Page 168
12.1 The knot cobordism group......Page 169
12.2 The matrix cobordism group......Page 170
12.3 Link cobordism......Page 177
Supplementary notes for Chapter 12......Page 183
13.1 A normal form......Page 184
13.2 Constructing 2-knots......Page 190
13.3 Seifert hypersurfaces......Page 192
13.4 Exteriors of 2-knots......Page 193
13.5 Cyclic covering spaces......Page 195
13.6 The k-invariant......Page 196
13.7 Ribbon presentations......Page 198
Supplementary notes for Chapter 13......Page 200
14.1 High-dimensional knot groups......Page 201
14.2 Ribbon 2-knot groups......Page 205
14.3 Torsion elements and the deficiency of 2-knot groups......Page 208
Supplementary notes for Chapter 14......Page 211
15.1 Topology of molecules......Page 213
15.2 Uses of the notion of equivalence......Page 214
15.3 Uses of the notion of neighborhood-equivalence......Page 217
Supplementary notes for Chapter 15......Page 220
16.1 Vassiliev-Gusarov algebra......Page 221
16.2 Vassiliev-Gusarov invariants and Jones type polynomials......Page 223
16.3 Kontsevich\'s iterated integral invariant......Page 226
16.4 Numerical invariants not of Vassiliev-Gusarov type......Page 229
Supplementary notes for Chapter 16......Page 231
Appendix A The equivalence of several notions of \"link equivalence\"......Page 232
B.1 The fundamental group......Page 235
B.2 Definitions and properties of covering spaces......Page 236
B.3 The classification of covering spaces......Page 238
B.4 Covering transformations and the monodromy map......Page 239
B.5 Branched covering spaces......Page 240
C.1 The connected sum......Page 242
C.3 The equivariant loop and sphere theorems......Page 243
C.4 Haken manifolds......Page 244
C.5 Seifert manifolds......Page 245
C.6 The annulus and torus theorems and the torus decomposition theorem......Page 247
C.7 Hyperbolic 3-manifolds......Page 248
D.1 Heegaard splittings......Page 250
D.2 Dehn surgery descriptions......Page 252
Appendix E The Blanchfield duality theorem......Page 255
F.0 Comments on the data......Page 260
F.1 Knot diagrams......Page 262
F.2 Type and symmetry......Page 270
F.3 Knot invariants......Page 277
F.4 Presentation matrices......Page 284
F.5 Ribbon presentations......Page 286
F.6 Skein and Kauffman polynomials......Page 289
F.7 Surface-link diagrams......Page 310
References......Page 312
Index......Page 421