Title: The geometry of unknotting tunnels
Abstract: Given a knot K in S^3, an unknotting tunnel for K is an arc τ from K to K, such that the complement of K and τ is a genus-2 handlebody. Fifteen years ago, Colin Adams asked a series of questions about how unknotting tunnels fit into the hyperbolic structure on the knot complement. For example: is τ isotopic to a geodesic? Can it be arbitrarily long, relative to a maximal cusp neighborhood? Does τ appear as an edge in the canonical polyhedral decomposition?
Although the most general versions of these questions are still open today, I will describe fairly complete answers in the case where K is created by a “generic” Dehn filling. As an application, there is an explicit family of knots in S^3 whose tunnels are arbitrarily long. This is joint work with Daryl Cooper and Jessica Purcell.