The geometry of unknotting tunnels

David Futer, Temple University

Abstract: Let K be a knot that lives in a closed manifold such as S^3. Then an unknotting tunnel for K is an arc \tau running from K to K, with the property that the complement of K and \tau is a handlebody. Equivalently, the unknotting tunnel defines a genus-2 Heegaard splitting of the complement of K.

In the generic situation where the complement of K is hyperbolic, we can ask a number of geometric questions about the tunnel. The following questions have been open since the mid 1990s: Is \tau isotopic to a geodesic? Is it isotopic to an edge of the canonical triangulation? Can \tau be arbitrarily long?

I will present some recent joint work, joint with Jessica Purcell, that answers these 3 questions if the knot K is created by long Dehn filling.