Dehn filling and the geometry of unknotting tunnels
arXiv:1105.3461 · doi:10.2140/gt.2013.17.1815
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.
45 pages, 17 figures. v3 contains minor revisions. To appear in Geometry & Topology
References in corpus (7)
- Random Heegaard splittings
- Rigidity of polyhedral surfaces, II
- A random tunnel number one 3-manifold does not fiber over the circle
- Canonical triangulations of Dehn fillings
- Proving a manifold to be hyperbolic once it has been approximated to be so
- Explicit Dehn filling and Heegaard splittings
- The length of unknotting tunnels