Scott A. Taylor

Mathematics and Statistics Department,
Colby College,
Waterville, ME 04901, USA .

Title: Exceptional Surgeries & Bridge Distance I: The role of thin position

Abstract: In this first of three talks, I will describe two fundamental questions in the theory of Dehn surgery on knots in 3-manifolds: (A) What surgeries on hyperbolic knots in hyperbolic 3-manifolds produce non-hyperbolic 3-manifolds? an (B) What surgeries on hyperbolic knots in hyperbolic 3-manifolds produce homeomorphic 3-manifolds? It turns out that the theory of Heegaard splittings and thin position can be used to give simple combinatorial obstructions to such exceptional surgeries. This introductory talk should be accessible to those with only a small bit of background in 3-manifold theory.
All three talks present joint work with Ryan Blair, Marion Campisi, Jesse Johnson, and Maggy Tomova.

Title: Exceptional Surgeries & Bridge Distance II: The genus of an alternately sloped essential surface bounds distance

Abstract: This talk will develop the ideas introduced in the first lecture and will show how the genus (and not just the Euler characteristic) of a surface in the complement of the knot can be used to give an upper bound on the so-called "bridge distance" of the knot. The talk will conclude with a proof that knots of bridge distance at least 3 have no surgeries producing a reducible 3-manifold. This shows, in particular, that knots of bridge distance at least 3 satisfy the Cabling Conjecture of Gonzalez-Acuña and Short.

Title: Exceptional Surgeries & Bridge Distance III: The genus of an alternately sloped bridge surface bounds distance

Abstract: This final talk will define the notions of spanning and splitting for bridge surfaces (ideas originally introduced in the context of Heegaard splittings by Johnson) and will show how they can be used to show that the genus of a bridge surface for the core of the surgery solid torus can be used to give an upper bound on the bridge distance of a knot. The talk will conclude with a proof that a hyperbolic knot in a hyperbolic 3-manifold having a non-hyperbolic surgery has bridge distance at most 6.