# [Topology and Geometry Seminar] “Cusps and alternating knots" by Prof Jessica Purcell (Monash University)

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The seminar aims to introduce research topics in topology, geometry and its interactions with other sciences. Anyone interested in mathematics is welcome to attend.

This week, Professor Jessica Purcell (Monash University) will discuss "Cusps and alternating knots".

**Abstract:** Alternating knots are some of the simplest knots to describe, and they occur frequently in low crossing knot tables. Most alternating knots have a complement that admits a hyperbolic metric: a metric with constant curvature -1. However, it is difficult to relate the hyperbolic geometry of these knots to their diagrams, and there are several open conjectures on possible relationships. In this talk, we will address one such conjecture, concerning cusp volume. We will define the cusp volume, show examples, and discuss recent results showing 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. This work is joint with M. Lackenby.

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