Chapter 1 - Hyperbolic Knots

Abstract

In view of the Mostow rigidity theorem, when the volume associated with the manifold is finite, the hyperbolic metrics are unique. Hence, geometric invariants coming out of the hyperbolic structure can be utilized to potentially distinguish between manifolds. It is to be noted that the complement is a finite volume but noncompact hyperbolic 3-manifold. This chapter provides a description of some of the applications, including the Smith Conjecture and the determination of symmetries of a knot. In case a link is found, it is possible to expand the cusps until they touch each other or themselves, to obtain a maximal set of cusps. In the case that the limit set is an actual circle, and the planes are geodesic, it is said that the surface is Fuchsian or totally geodesic. For instance, in [AS03], it is proved that 2-bridge knots never have totally geodesic Seifert surfaces. An interesting question is whether there are totally geodesic surfaces in knot complementing other than such Seifert surfaces. This is one of many open questions that still remain in the theory of hyperbolic knots.