A quantum obstruction to purely cosmetic surgeries

Abstract

We present new obstructions for a knot K in S^3 to admit purely cosmetic surgeries, which arise from the study of Witten-Reshetikhin-Turaev invariants at fixed level. In particular, we strengthen a recent result of Hanselman, showing that if K has purely cosmetic surgeries then the slopes of the surgeries are of the form 1/5k except if the Jones polynomial of K evaluated at a 5-th root of unity is 1. For any odd prime r, we also give an obstruction for K to have a 1/k surgery slope with k coprime to r that involves the values of the first (r-3)/2 colored Jones polynomials of K at an r-th root of unity. We verify the purely cosmetic surgery conjecture for all knots with at most 17 crossings.