Knots and links

Contents......Page 3
Preface......Page 8
Notation......Page 12
1.1 Knots......Page 15
1.2 Deformation......Page 17
1.3 Polygonal knots......Page 19
1.4 Smooth knots......Page 22
1.5 Torus knots......Page 23
1.6 Fourier knots and Lissajous knots......Page 25
1.7 Topological symmetries......Page 27
1.8 Links......Page 30
1.9 Invariants......Page 34
1.10 The creation of catalogues......Page 35
1.11 A technicality......Page 38
1.12 Exercises......Page 43
2 A Topologist's Toolkit......Page 45
2.1 The 3-sphere......Page 46
2.2 Manifolds......Page 48
2.3 Orientation and orientability......Page 49
2.4 Separation and bounding......Page 50
2.6 Surfaces......Page 52
2.7 Euler characteristic and genus......Page 55
2.8 Surgery......Page 56
2.9 Compressibility......Page 57
2.10 Transverse intersections and general position......Page 58
2.11 Graphs......Page 59
2.12 The combinatorial approach......Page 62
2.13 Exercises......Page 64
3.2 Projections......Page 65
3.3 Diagrams......Page 67
3.4 Some families of links......Page 70
3.5 Crossing number......Page 71
3.6 Invariants from oriented diagrams......Page 76
3.7 A warning about minimal diagrams......Page 78
3.8 Moves on diagrams......Page 80
3.9 The classification problem......Page 86
3.10 Exercises......Page 90
4.2 Satellites and companionship......Page 92
4.3 Factorising links......Page 95
4.4 Prime satellites......Page 97
4.5 Uniqueness of factorisation......Page 99
4.6 The product operation......Page 102
4.7 Link invariants and factorisation......Page 104
4.8 Hyperbolic links......Page 107
4.9 Tangle decomposition......Page 109
4.10 Bridge presentations......Page 112
4.11 Exercises......Page 114
5.1 Seifert's algorithm......Page 116
5.2 Seifert graphs......Page 119
5.3 Genus-1 knots......Page 120
5.4 Different genus measures......Page 124
5.5 Surgery equivalence......Page 128
5.6 Genus and factorisation......Page 132
5.7 Linking number revisited......Page 135
5.8 Genus of satellites......Page 138
5.9 Exercises......Page 141
6.1 Loops in graphs......Page 143
6.2 Simplicial complexes......Page 146
6.3 Loops in simplicial complexes......Page 147
6.4 Homology of surfaces......Page 151
6.5 The Seifert matrix......Page 153
6.6 Link invariants from the Seifert matrix......Page 156
6.7 Properties of signature and determinant......Page 159
6.8 Signature and unknotting number......Page 165
6.9 Exercises......Page 168
7.1 The Alexander polynomial......Page 171
7.2 The Alexander polynomial and genus......Page 175
7.3 The Alexander skein relation......Page 176
7.4 The Conway polynomial......Page 177
7.5 Resolving trees......Page 181
7.6 Homogeneous links......Page 184
7.7 Linear skein theory......Page 188
7.8 Marked tangles and mutation......Page 191
7.9 Conway's fraction formula......Page 195
7.10 Cylindrical tangles......Page 197
7.11 A troublesome pair......Page 199
7.12 Exercises......Page 200
8.1 Generating rational tangles......Page 203
8.2 Classification of rational tangles......Page 207
8.3 Homology and maps......Page 210
8.4 Automorphisms of the torus......Page 212
8.5 Representing tangle operations by matrices......Page 213
8.6 Continued fractions......Page 216
8.7 Rational links......Page 219
8.8 Enumerating rational links......Page 223
8.9 Applications of rational tangles......Page 225
8.10 Exercises......Page 228
9.1 Discovery of the Jones polynomial......Page 229
9.2 The Kauffman bracket......Page 231
9.3 Properties of the bracket polynomial......Page 235
9.4 States of graphs and diagrams......Page 238
9.5 Adequate diagrams......Page 243
9.6 Bounds on crossing number......Page 249
9.7 Polynomial link invariants......Page 250
9.8 Exercises......Page 253
10.1 Braid presentations......Page 255
10.2 The Homfly polynomial......Page 258
10.3 The Homfly polynomial and braid index......Page 261
10.4 Braid index of rational links......Page 263
10.5 Classification of torus knots......Page 267
10.6 Arc presentations......Page 270
10.7 Arc presentations on diagrams......Page 275
10.8 Arc presentations from diagrams......Page 280
10.9 Braid presentations of satellites......Page 286
10.10 Applications of the open-book infrastructure......Page 294
10.11 Exercises......Page 298
Appendix A Knot Diagrams......Page 300
Appendix B Numerical Invariants......Page 308
Appendix C Properties......Page 311
Appendix D Polynomials......Page 314
Appendix E Polygon Coordinates......Page 320
Appendix F Family Properties......Page 321
Bibliography......Page 322
Index......Page 337