Characterizing slopes for satellite knots

A slope $p\mathrm{/}q$ is said to be characterizing for a knot $K$ if the homeomorphism type of the $p\mathrm{/}q$-Dehn surgery along $K$ determines the knot up to isotopy. Extending previous work of Lackenby and McCoy on hyperbolic and torus knots respectively, we study satellite knots to show that for a knot $K$, any slope $p\mathrm{/}q$ is characterizing provided $\mathrm{\mid }q\mathrm{\mid }$ is sufficiently large. In particular, we establish that every non-integral slope is characterizing for a composite knot. Our approach consists of a detailed examination of the JSJ decomposition of a surgery along a knot, combined with results from other authors giving constraints on surgery slopes that yield manifolds containing certain surfaces.