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

TL;DR

This paper investigates constraints on knots in three-dimensional space that can admit chirally cosmetic surgeries, using various invariants, and shows most knots do not admit such surgeries.

Contribution

It introduces new constraints based on quantum invariants and Floer homology to identify knots that cannot admit chirally cosmetic surgeries.

Findings

Approximately 75% of certain knots admit no chirally cosmetic surgeries.

Constraints derived exclude many knots from admitting such surgeries.

Results apply to knots with up to 10 crossings, excluding amphicheiral and certain torus knots.

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 $SO(3)$-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.