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.
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