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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Update: The book is available for pre-order from the AMS Bookstore:
Hyperbolic Knot Theory

Or a draft version is available on the arXiv. The arxiv version has a few known errors that should be fixed in the published version.