Title: Cusp volumes of alternating knots
Speaker: Jessica Purcell (Brigham Young University/University of Melbourne)
Abstract: Alternating knots are some of the simplest knots to describe, and they occur frequently in low crossing knot tables. While most alternating knots are hyperbolic, it is still difficult to relate the hyperbolic geometry of these knots to their diagrams. Based on a large amount of experimental evidence, there are several open conjectures relating the hyperbolic geometry of alternating knots to their diagrammatical properties. In this talk, we will address one such conjecture, concerning cusp volume. We will define the cusp volume, and show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. In addition to giving diagrammatical estimates on cusp volume, this also leads to geometric estimates on lengths of slopes in terms of a diagram of the knot. All these estimates are explicit. This is joint work with Marc Lackenby.