Hyperbolic Knot Theory and Geometric Triangulations

Abstract

We begin by studying hyperbolic geometry and hyperbolic structures on manifolds, looking at classical examples of hyperbolic manifolds and some important results on their structure and rigidity. We study hyperbolic knot complements, starting with methods to triangulate knot complements. We see how a triangulation by geometric simplices can give rise to a geometric structure on a manifold using Thurston's gluing and completeness equations. The structure of the various parts of a hyperbolic manifold is given by the Margulis theorem. Finally, we study the equivalence problem for knots in the 3-sphere and the homeomorphism problem for hyperbolic 3-manifolds using geometric triangulations.