Dehn filling and the geometry of unknotting tunnels

Abstract

Any one-cusped hyperbolic manifold M with an unknotting tunnel x is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that r is isotopic to a geodesic, and characterize whether r is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of r relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma and Weeks. We also construct an explicit sequence of one-tunnel knots in S3, all of whose unknotting tunnels have length approaching infinity.