Characterising slopes for knots of hyperbolic type

A slope p/q is characterising for a knot K in the 3-sphere if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. Sorya showed that for any knot K, there exists a constant C(K) such that any slope p/q with |q|≥C(K) is characterising for K. However, the proof of the existence of C(K) in the general case is non-constructive, which naturally evokes the question of how to compute explicit values for C(K). In this talk, I will explore methods for finding C(K) in the case where K is a knot of hyperbolic type (meaning that the JSJ decomposition of its complement has a hyperbolic outermost JSJ piece). I will begin with the simplest case, in which K is a hyperbolic knot; time permitting, I will also discuss some ongoing joint work with Patricia Sorya on the more general case.