Dehn filling and the geometry of unknotting tunnels

Cooper, Daryl¹ ; Futer, David² ; Purcell, Jessica S³

¹ Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106, USA
² Department of Mathematics, Temple University, Philadelphia, PA 19147, USA
³ Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA

Geometry & topology, Tome 17 (2013) no. 3, pp. 1815-1876.

Résumé

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.

Détail

DOI : 10.2140/gt.2013.17.1815

Classification : 57M25, 57M50, 57R52
Keywords: unknotting tunnel, hyperbolic 3–manifold, hyperbolic knot, geodesic, length, Dehn filling

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