On constraints for knots to admit chirally cosmetic surgeries and their calculations

Kazuhiro Ichihara, Tetsuya Ito and Toshio Saito

Vol. 321 (2022), No. 1, 167–191

DOI: 10.2140/pjm.2022.321.167

Abstract

We discuss various constraints for knots in $S^3$ to admit chirally cosmetic surgeries, derived from invariants of 3-manifolds, such as, the quantum $\mathrm{SO}<!--FUNCTION\; APPLICATION-->(3)$-invariant, the rank of the Heegaard Floer homology, and finite-type invariants. We apply them to show that a large portion (roughly 75%) of knots which are neither amphicheiral nor $(2,p)$-torus knots with less than or equal to 10 crossings admits no chirally cosmetic surgeries.

Dedicated to Professor Kimihiko Motegi on his sixtieth birthday

Keywords

chirally cosmetic surgery

Mathematical Subject Classification

Primary: 57K10, 57K16, 57K18, 57K31

Milestones

Received: 7 April 2022

Revised: 14 July 2022

Accepted: 24 July 2022

Published: 3 March 2023