Hyperbolic Knot Theory

TL;DR

This book introduces hyperbolic geometry in three dimensions and explores its applications to knot theory, focusing on geometric structures, specific knot families, and invariants like volume and the A-polynomial.

Contribution

It provides a comprehensive introduction to hyperbolic geometry in 3D and applies it to study various knot families and their geometric invariants.

Findings

Analysis of hyperbolic structures on knots and links

Development of geometric techniques like angle structures

Insights into hyperbolic invariants such as volume and A-polynomial

Abstract

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.