This book provides an introduction to hyperbolic geometry in dimension three, with motivation and applications arising from knot theory. Hyperbolic geometry was first used as a tool to study knots by Riley and then Thurston in the 1970s. By the 1980s, combining work of Mostow and Prasad with Gordon and Luecke, it was known that a hyperbolic structure on a knot complement in the 3-sphere gives a complete knot invariant. However, it remains a difficult problem to relate the hyperbolic geometry of a knot to other invariants arising from knot theory. In particular, it is difficult to determine hyperbolic geometric information from a knot diagram, which is classically used to describe a knot. This textbook provides background on these problems, and tools to determine hyperbolic information on knots. It also includes results and state-of-the art techniques on hyperbolic geometry and knot theory to date.
The book was written to be interactive, with many examples and exercises. Some important results are left to guided exercises. The level is appropriate for graduate students with a basic background in algebraic topology, particularly fundamental groups and covering spaces. Some experience with some differential topology and Riemannian geometry will also be helpful.
Author(s): Jessica S. Purcell
Series: Graduate Studies in Mathematics 209
Edition: 1
Publisher: American Mathematical Society
Year: 2020
Language: English
Pages: 369
City: Providence
Tags: Hyperbolic Geometry, Knots
Contents 8
Preface 12
Why I wrote this book 12
How I structured the book 13
Prerequisites and notes to students 15
Acknowledgments 15
Introduction 18
Chapter 0. A Brief Introduction to Hyperbolic Knots 20
0.1. An introduction to knot theory 20
0.2. Problems in knot theory 23
0.3. Exercises 35
Part 1 . Foundations of Hyperbolic Structures 36
Chapter 1. Decomposition of the Figure-8 Knot 38
1.1. Polyhedra 38
1.2. Generalizing: Exercises 45
Chapter 2. Calculating in Hyperbolic Space 48
2.1. Hyperbolic geometry in dimension two 48
2.2. Hyperbolic geometry in dimension three 57
2.3. Exercises 59
Chapter 3. Geometric Structures on Manifolds 64
3.1. Geometric structures 64
3.2. Complete structures 71
3.3. Developing map and completeness 82
3.4. Exercises 83
Chapter 4. Hyperbolic Structures and Triangulations 86
4.1. Geometric triangulations 86
4.2. Edge gluing equations 90
4.3. Completeness equations 96
4.4. Computing hyperbolic structures 100
4.5. Exercises 101
Chapter 5. Discrete Groups and the Thick-Thin Decomposition 104
5.1. Discrete subgroups of hyperbolic isometries 104
5.2. Elementary groups 110
5.3. Thick and thin parts 113
5.4. Hyperbolic manifolds with finite volume 116
5.5. Universal elementary neighborhoods 118
5.6. Exercises 125
Chapter 6. Completion and Dehn Filling 128
6.1. Mostow–Prasad rigidity 128
6.2. Completion of incomplete structures 129
6.3. Hyperbolic Dehn filling space 133
6.4. A brief summary of geometric convergence 141
6.5. Exercises 147
Part 2 . Tools, Techniques, and Families of Examples 150
Chapter 7. Twist Knots and Augmented Links 152
7.1. Twist knots and Dehn fillings 152
7.2. Double twist knots and the Borromean rings 157
7.3. Augmenting and highly twisted knots 160
7.4. Cusps of fully augmented links 166
7.5. Exercises 172
Chapter 8. Essential Surfaces 176
8.1. Incompressible surfaces 176
8.2. Torus decomposition, Seifert fibering, and geometrization 182
8.3. Normal surfaces, angled polyhedra, and hyperbolicity 184
8.4. Pleated surfaces and a 6-theorem 192
8.5. Exercises 200
Chapter 9. Volume and Angle Structures 202
9.1. Hyperbolic volume of ideal tetrahedra 202
9.2. Angle structures and the volume functional 212
9.3. Leading-trailing deformations 214
9.4. The Schläfli formula 220
9.5. Consequences 221
9.6. Exercises 223
Chapter 10. Two-Bridge Knots and Links 226
10.1. Rational tangles and 2-bridge links 226
10.2. Triangulations of 2-bridge links 230
10.3. Positively oriented tetrahedra 239
10.4. Maximum in interior 246
10.5. Exercises 255
Chapter 11. Alternating Knots and Links 258
11.1. Alternating diagrams and hyperbolicity 259
11.2. Checkerboard surfaces 272
11.3. Exercises 276
Chapter 12. The Geometry of Embedded Surfaces 278
12.1. Belted sums and mutations 279
12.2. Fuchsian, quasifuchsian, and accidental surfaces 283
12.3. Fibers and semifibers 289
12.4. Exercises 295
Part 3 . Hyperbolic Knot Invariants 298
Chapter 13. Estimating Volume 300
13.1. Summary of bounds encountered so far 300
13.2. Negatively curved metrics and Dehn filling 304
13.3. Volume, guts, and essential surfaces 318
13.4. Exercises 327
Chapter 14. Ford Domains and Canonical Polyhedra 330
14.1. Horoballs and isometric spheres 331
14.2. Ford domain 338
14.3. Canonical polyhedra 345
14.4. Exercises 350
Chapter 15. Algebraic Sets and the ?-Polynomial 352
15.1. The gluing variety 352
15.2. Representations of knots 358
15.3. The ?-polynomial 366
15.4. Exercises 370
Bibliography 372
Index 382
