Hyperbolic Volumes of Two Bridge Cone-Manifolds

Abstract

In this paper we investigate the existence of hyperbolic, Euclidean and spherical structures on cone-manifolds with underlying space 3-sphere and with singular set a given two-bridge knot. For two-bridge knots with 8 crossings we present trigonometric identities involving the length of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic space.

Keywords

• cone-manifold
• orbifold
• two-bridge knot
• volume
• geodesic length