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.

Review

There are many existing books on hyperbolic geometry and on knot theory taken separately, but to my knowledge, this is the first that substantially focuses on the two fields together. The combination benefits each of the constituents. This book will be useful both as an introduction and as a reference for those interested in either (or both!) topics. --Henry Segerman, Oklahoma State University

Table of Contents

Chapter 0. A Brief Introduction to Hyperbolic Knots

0.1. An introduction to knot theory

0.2. Problems in knot theory

0.3. Exercises

Cart 1 . Foundations of Hyperbolic Structures

Chapter 1. Decomposition of the Figure-8 Knot

Chapter 2. Calculating in Hyperbolic Space

Chapter 3. Geometric Structures on Manifolds

Chapter 4. Hyperbolic Structures and Triangulations

Chapter 5. Discrete Groups and the Thick-Thin Decomposition

Chapter 6. Completion and Dehn Filling

Cart 2. Tools, Techniques, and Families of Examples

Chapter 7. Twist Knots and Augmented links

Chapter 8. Essential Surfaces

Chapter 9. Volume and Angle Structures

Chapter 10. Two-Bridge Knots and links

Chapter 11. Alternating Knots and links

Chapter 12. The Geometry of Embedded Surfaces

Cart 3 . Hyperbolic Knot Invariants

Chapter 13. Estimating Volume

Chapter 14. Ford Domains and Canonical Polyhedra

Chapter 15. Algebraic Sets and the A-Polynomial