Dehn filling and the geometry of unknotting tunnels

Abstract

Any one-cusped hyperbolic manifold $M$ with an unknotting tunnel $\tau$ 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 $\tau$ is isotopic to a geodesic, and characterize whether $\tau$ is isotopic to an edge in the canonical decomposition of $M$. We also give explicit estimates (with additive error only) on the length of $\tau$ 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 $S^3$, all of whose unknotting tunnels have length approaching infinity.