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Dehn filling and the
geometry of unknotting tunnels
Daryl Cooper, David Futer and Jessica S Purcell |
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Geometry & Topology 17 (2013) 1815–1876
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Abstract |
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Any one-cusped hyperbolic manifold with an unknotting tunnel is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where is obtained by “generic” Dehn filling, we prove that is isotopic to a geodesic, and characterize whether is isotopic to an edge in the canonical decomposition of . We also give explicit estimates (with additive error only) on the length of 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 , all of whose unknotting tunnels have length approaching infinity. |
Keywords
unknotting tunnel, hyperbolic 3–manifold, hyperbolic knot,
geodesic, length, Dehn filling
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Mathematical Subject Classification 2010
Primary: 57M25, 57M50, 57R52
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References |
Publication
Received: 13 August 2012
Accepted: 8 March 2013
Published: 10 July 2013
Proposed: Walter Neumann
Seconded: Danny Calegari, Colin Rourke
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Authors |
| Daryl Cooper | |
| Department of Mathematics University of California, Santa Barbara Santa Barbara, CA 93106 USA |
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| http://www.math.ucsb.edu/~cooper/ | |
| David Futer | |
| Department of Mathematics Temple University Philadelphia, PA 19147 USA |
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| http://math.temple.edu/~dfuter | |
| Jessica S Purcell | |
| Department of Mathematics Brigham Young University Provo, UT 84602-6539 USA |
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| http://www.math.byu.edu/~jpurcell/ | |
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