Hyperbolic Knot Theory

This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.

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Newest Update: 28 April 2020.

Download the book: HypKnotTheory.pdf or arXiv

Contents:

  • Front Matter: Preface, Table of Contents
  1. A Brief Introduction to Hyperbolic Knots
Part 1. Foundations of Hyperbolic Structures
  1. Decomposition of the Figure-8 Knot
  2. Calculating in Hyperbolic Space
  3. Geometric Structures on Manifolds
  4. Hyperbolic Structures and Triangulations
  5. Discrete Groups and the Thick-Thin Decomposition
  6. Completion and Dehn Filling
Part 2. Tools, Techniques, and Families of Examples
  1. Twist Knots and Augmented Links
  2. Essential Surfaces
  3. Volume and Angle Structures
  4. Two-Bridge Knots and Links
  5. Alternating Knots and Links
  6. The Geometry of Embedded Surfaces
Part 3. Hyperbolic Knot Invariants.
  1. Estimating Volume
  2. Ford Domains and Canonical Polyhedra
  3. Representations and the A-Polynomial
  • Bibliography
  • Index