If you are a student of mathematics, a scientist working in fields affected by knot theory research, or a curious amateur who finds mathematics intriguing, The Knot Book is for you. With this engagingly written and illustrated book, you will be working with some of the most advanced ideas in contemporary mathematics.
Author(s): Colin Conrad Adams
Publisher: W.H. Freeman & Company
Year: 1994
Language: English
Pages: 308
Contents......Page 2
Preface......Page 5
1.1 Introduction......Page 8
1.2 Composition of Knots......Page 14
1.3 Reidemeister Moves......Page 19
1.4 Links......Page 23
1.5 Tricolorability......Page 29
1.6 Knots and Sticks......Page 34
2.1 History of Knot Tabulation......Page 38
2.2 The Dowker Notation for Knots......Page 42
2.3 Conway's Notation......Page 48
2.4 Knots and Planar Graphs......Page 58
3.1 Unknotting Number......Page 63
3.2 Bridge Number......Page 70
3.3 Crossing Number......Page 73
4.1 Surfaces without Boundary......Page 76
4.2 Surfaces with Boundary......Page 92
4.3 Genus and Seifert Surfaces......Page 100
5.1 Torus Knots......Page 112
5.2 Satellite Knots......Page 120
5.3 Hyperbolic Knots......Page 124
5.4 Braids......Page 132
5.5 Almost Alternating Knots......Page 144
6.1 The Bracket Polynomial and the Jones Polynomial......Page 152
6.2 Polynomials of Alternating Knots......Page 161
6.3 The Alexander and HOMELY Polynomials......Page 170
6.4 Amphicheirality......Page 181
7.1 DNA......Page 185
7.2 Synthesis of Knotted Molecules......Page 199
7.3 Chirality of Molecules......Page 205
7.4 Statistical Mechanics and Knots......Page 209
8.1 Links in Graphs......Page 218
8.2 Knots in Graphs......Page 225
8.3 Polynomials of Graphs......Page 234
9.1 Knot Complements and Three-Manifolds......Page 245
9.2 The Three-Sphere and Lens Spaces......Page 248
9.3 The Poincaré Conjecture, Dehn Surgery, and the Gordon-Luecke Theorem......Page 259
10.1 Picturing Four Dimensions......Page 267
10.2 Knotted Spheres in Four Dimensions......Page 274
10.3 Knotted Three-Spheres in Five-Space......Page 275
Knot Jokes and Pastimes......Page 278
Appendix: Table of Knots, Links, and Knot and Link Invariants......Page 281
Suggested Readings and References......Page 293
Index......Page 305
