#### Volume 17, issue 3 (2013)

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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
##### 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.

##### Keywords
unknotting tunnel, hyperbolic 3–manifold, hyperbolic knot, geodesic, length, Dehn filling
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M50, 57R52
##### Publication
Accepted: 8 March 2013
Published: 10 July 2013
Proposed: Walter Neumann
Seconded: Danny Calegari, Colin Rourke
##### Authors
 Daryl Cooper Department of Mathematics University of California, Santa Barbara Santa Barbara, CA 93106 USA http://www.math.ucsb.edu/~cooper/ David Futer Department of Mathematics Temple University Philadelphia, PA 19147 USA http://math.temple.edu/~dfuter Jessica S Purcell Department of Mathematics Brigham Young University Provo, UT 84602-6539 USA http://www.math.byu.edu/~jpurcell/