Hyperbolic Knot Theory

Contents
Preface
Why I wrote this book
How I structured the book
Prerequisites and notes to students
Acknowledgments
Introduction
Chapter 0. A Brief Introduction to Hyperbolic Knots
0.1. An introduction to knot theory
0.2. Problems in knot theory
0.3. Exercises
Part 1 . Foundations of Hyperbolic Structures
Chapter 1. Decomposition of the Figure-8 Knot
1.1. Polyhedra
1.2. Generalizing: Exercises
Chapter 2. Calculating in Hyperbolic Space
2.1. Hyperbolic geometry in dimension two
2.2. Hyperbolic geometry in dimension three
2.3. Exercises
Chapter 3. Geometric Structures on Manifolds
3.1. Geometric structures
3.2. Complete structures
3.3. Developing map and completeness
3.4. Exercises
Chapter 4. Hyperbolic Structures and Triangulations
4.1. Geometric triangulations
4.2. Edge gluing equations
4.3. Completeness equations
4.4. Computing hyperbolic structures
4.5. Exercises
Chapter 5. Discrete Groups and the Thick-Thin Decomposition
5.1. Discrete subgroups of hyperbolic isometries
5.2. Elementary groups
5.3. Thick and thin parts
5.4. Hyperbolic manifolds with finite volume
5.5. Universal elementary neighborhoods
5.6. Exercises
Chapter 6. Completion and Dehn Filling
6.1. Mostow–Prasad rigidity
6.2. Completion of incomplete structures
6.3. Hyperbolic Dehn filling space
6.4. A brief summary of geometric convergence
6.5. Exercises
Part 2 . Tools, Techniques, and Families of Examples
Chapter 7. Twist Knots and Augmented Links
7.1. Twist knots and Dehn fillings
7.2. Double twist knots and the Borromean rings
7.3. Augmenting and highly twisted knots
7.4. Cusps of fully augmented links
7.5. Exercises
Chapter 8. Essential Surfaces
8.1. Incompressible surfaces
8.2. Torus decomposition, Seifert fibering, and geometrization
8.3. Normal surfaces, angled polyhedra, and hyperbolicity
8.4. Pleated surfaces and a 6-theorem
8.5. Exercises
Chapter 9. Volume and Angle Structures
9.1. Hyperbolic volume of ideal tetrahedra
9.2. Angle structures and the volume functional
9.3. Leading-trailing deformations
9.4. The Schläfli formula
9.5. Consequences
9.6. Exercises
Chapter 10. Two-Bridge Knots and Links
10.1. Rational tangles and 2-bridge links
10.2. Triangulations of 2-bridge links
10.3. Positively oriented tetrahedra
10.4. Maximum in interior
10.5. Exercises
Chapter 11. Alternating Knots and Links
11.1. Alternating diagrams and hyperbolicity
11.2. Checkerboard surfaces
11.3. Exercises
Chapter 12. The Geometry of Embedded Surfaces
12.1. Belted sums and mutations
12.2. Fuchsian, quasifuchsian, and accidental surfaces
12.3. Fibers and semifibers
12.4. Exercises
Part 3 . Hyperbolic Knot Invariants
Chapter 13. Estimating Volume
13.1. Summary of bounds encountered so far
13.2. Negatively curved metrics and Dehn filling
13.3. Volume, guts, and essential surfaces
13.4. Exercises
Chapter 14. Ford Domains and Canonical Polyhedra
14.1. Horoballs and isometric spheres
14.2. Ford domain
14.3. Canonical polyhedra
14.4. Exercises
Chapter 15. Algebraic Sets and the ?-Polynomial
15.1. The gluing variety
15.2. Representations of knots
15.3. The ?-polynomial
15.4. Exercises
Bibliography
Index