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Dehn filling and the geometry of unknotting tunnels

Daryl Cooper, David Futer and Jessica S Purcell

Geometry & Topology 17 (2013) 1815–1876

Any one-cusped hyperbolic manifold M with an unknotting tunnel τ 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 τ is isotopic to a geodesic, and characterize whether τ is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of τ 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.

unknotting tunnel, hyperbolic 3–manifold, hyperbolic knot, geodesic, length, Dehn filling
Mathematical Subject Classification 2010
Primary: 57M25, 57M50, 57R52
Received: 13 August 2012
Accepted: 8 March 2013
Published: 10 July 2013
Proposed: Walter Neumann
Seconded: Danny Calegari, Colin Rourke
Daryl Cooper
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106
David Futer
Department of Mathematics
Temple University
Philadelphia, PA 19147
Jessica S Purcell
Department of Mathematics
Brigham Young University
Provo, UT 84602-6539