Dehn filling and the geometry of unknotting tunnels

TL;DR

This paper investigates the geometry of unknotting tunnels in hyperbolic 3-manifolds, proving they are geodesic under generic Dehn fillings, characterizing their isotopy classes, and providing explicit length estimates, thus addressing longstanding questions.

Contribution

It establishes that unknotting tunnels are geodesic in generic cases, characterizes their isotopy to edges, and offers explicit length bounds, advancing understanding of hyperbolic 3-manifold topology.

Findings

Unknotting tunnels are isotopic to geodesics in generic Dehn fillings.

Characterization of tunnels as isotopic to edges in canonical decompositions.

Explicit length estimates for tunnels relative to maximal cusps.

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.