Hyperbolic Knot Theory

Abstract

In this thesis, we study Thurston’s approach to finding complete hyperbolic structures on 3-manifolds using ideal triangulations. This approach involves solving a set of equations called the Thurston’s gluing equations. These equations are nonlinear and difficult to solve, so Casson and Rivin developed the method of angle structures through which they separated Thurston’s equations into a linear and a non-linear part and extracted geometric information from each part separately. We also study geometric triangulations of constant curvature manifolds and how they are related by Pachner moves. We specially focus on understanding geometric ideal triangulations of cusped hyperbolic 3-manifolds and prove that any two geometric ideal triangulations have a common geometric subdivision with a finite number of polytopes. As a result, geometric ideal triangulations of a cusped hyperbolic 3-manifold become related by geometric Pachner moves. Along the way, we will discuss some foundational results in the theory of 3-manifolds, triangulations and hyperbolic geometry which we require for studying the central topics in this thesis.