Two-bridge knots admit no purely cosmetic surgeries

Abstract

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce $3$–manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or $3$–manifolds; for knots, we study the signature and some finite type invariants, and for $3$–manifolds, we deploy the $SL\left(2,ℂ\right)$ Casson invariant.