**Graduate Studies in Mathematics**

Volume: 209;
2020;
369 pp;
Softcover

MSC: Primary 57;
Secondary 30

**Print ISBN: 978-1-4704-5499-9
Product Code: GSM/209**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

**Electronic ISBN: 978-1-4704-6211-6
Product Code: GSM/209.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

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#### Supplemental Materials

# Hyperbolic Knot Theory

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*Jessica S. Purcell*

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.

#### Readership

Graduate students interested in hyperbolic geometry and knot theory.

#### Reviews & Endorsements

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

# Table of Contents

## Hyperbolic Knot Theory

- Contents 88
- Preface 1212
- Introduction 1818
- Chapter 0. A Brief Introduction to Hyperbolic Knots 2020
- Part 1 . Foundations of Hyperbolic Structures 3636
- Chapter 1. Decomposition of the Figure-8 Knot 3838
- Chapter 2. Calculating in Hyperbolic Space 4848
- Chapter 3. Geometric Structures on Manifolds 6464
- Chapter 4. Hyperbolic Structures and Triangulations 8686
- Chapter 5. Discrete Groups and the Thick-Thin Decomposition 104104
- Chapter 6. Completion and Dehn Filling 128128

- Part 2 . Tools, Techniques, and Families of Examples 150150
- Part 3 . Hyperbolic Knot Invariants 298298