Dave Futer will be giving a talk at Yale on Monday, 11/29:

**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.

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