The Dehn surgery characterisation of Whitehead doubles

A slope p/q is said to be characterising for a knot K if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. In this talk, I will discuss how JSJ decompositions and hyperbolic techniques can be used to find characterising slopes for a special class of satellite knots which includes Whitehead doubles. I will also exhibit a family of pairs of satellite knots sharing a non-characterising slope of the form 1/q.