On the number of shared Dehn surgeries between two knots

Abstract

A folklore theorem states that for any pair of distinct knots in $S^3$, performing $p/q$-Dehn surgery on each knot yields orientation-preservingly homeomorphic manifolds for at most finitely many slopes $p/q$. In this paper, we provide a proof based on the JSJ decomposition of knot exteriors. In particular, for any given pair of distinct knots, it provides an effective bound on the maximal number of shared surgeries between the knots.