A quantum obstruction to purely cosmetic surgeries

TL;DR

This paper introduces new quantum invariants-based obstructions to purely cosmetic surgeries on knots, providing conditions on surgery slopes and Jones polynomial evaluations, and verifies the conjecture for knots with up to 17 crossings.

Contribution

It develops novel quantum invariant obstructions for cosmetic surgeries and extends the verification of the conjecture to all knots with up to 17 crossings.

Findings

Purely cosmetic surgeries have slopes of the form 1/5k unless certain Jones polynomial conditions hold.

Obstructions involving colored Jones polynomials at roots of unity prevent certain surgery slopes.

The conjecture is verified for all knots with at most 17 crossings.

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.