A note on Jones polynomial and cosmetic surgery

Abstract

We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K^{\prime \prime }(1)\ne 0$ or $V_K^{\prime \prime \prime }(1)\ne 0$. 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.

13 pages, 4 figures

References in corpus (3)

• Floer homology and knot complements
• Knot Floer homology and the four-ball genus
• Surgery formulae for finite type invariants of rational homology 3--spheres

Cited by in corpus (1)

• Cosmetic operations and Khovanov multicurves