Communications in Analysis and Geometry

Volume 27 (2019)

Number 5

A note on Jones polynomial and cosmetic surgery

Pages: 1087 – 1104



Kazuhiro Ichihara (Department of Mathematics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan)

Zhongtao Wu (Department of Mathematics, Chinese University of Hong Kong)


We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V^{\prime \prime}_K (1) \neq 0$ or $V^{\prime \prime \prime}_K (1) \neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than 11 crossings except for three 10-crossing knots and five 11-crossing knots. We also compute the finite type invariant of order $3$ for two-bridge knots and Whitehead doubles, from which we prove several nonexistence results of purely cosmetic surgery.

Received 4 December 2016

Accepted 23 May 2017

Published 12 November 2019