What is a hyperbolic knot?

Abstract: One of the major advances in modern knot theory is the result of William Thurston that classifies all knots as one of three types: a torus knot, a satellite knot, or a hyperbolic knot. When a knot is hyperbolic, we can apply tools and results from hyperbolic geometry to study it. But what is a hyperbolic knot?!

In the first half of this talk we will discuss some general knot theory, the upper half space model of hyperbolic space, and what makes a knot hyperbolic. In the second half we will carefully step through the decomposition of the figure-8 knot complement into two ideal tetrahedra and use this decomposition to prove that the figure-8 knot is hyperbolic.

algebraic geometrycombinatoricsdifferential geometrynumber theoryrepresentation theory

Audience: learners

What is ...? Seminar

Series comments: The ``What is ... ? Seminar'' (WiSe) is a weekly Zoom mathematics seminar. The goal is to explain interesting things to each other in a casual manner. The name is inspired by the What is ...? column of the Notices of the American Mathematical Society. See the website for more details. If you want to receive the zoom link for the talks please fill out the form linked on the website or contact Anna at a.puskas@uq.edu.au.