Toronto, December 2 - 5, 2022
I will talk about work Alan McLeay investigating the above seemingly innocent questions, and the more general version: Given two surfaces, when does there admit a finite-sheeted cover of one over the other? A complete answer is available if the two surfaces are of finite type. In the infinite-type world, the question is less innocent than one might expect.
A slope $p/q$ is said to be characterizing for a knot $K$ if the homeomorphism type of the $p/q$-Dehn surgery along $K$ determines the knot up to isotopy.
Previous work of Lackenby and McCoy gives a condition for $p/q$ to be characterizing for a hyperbolic or torus knot $K$. By studying the JSJ decomposition of knot exteriors, we extend this result to satellite knots and obtain a characterizing condition for any given knot $K$.