Heegaard Floer homology and chirally cosmetic surgeries

TL;DR

This paper uses Heegaard Floer homology to identify new obstructions to chirally cosmetic surgeries on knots, leading to classifications and exclusions for certain knot families.

Contribution

It introduces novel obstructions from Heegaard Floer homology, especially using immersed curve formulations, to study chirally cosmetic surgeries.

Findings

Classified chirally cosmetic surgeries on odd alternating pretzel knots.

Ruled out such surgeries for many Whitehead doubles.

Excluded cosmetic surgeries for L-space knots with opposite slope signs.

Abstract

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. We find new obstructions to the existence of such surgeries coming from Heegaard Floer homology; in particular, we make use of immersed curve formulations of knot Floer homology and the corresponding surgery formula. As an application, we completely classify chirallly cosmetic surgeries on odd alternating pretzel knots, and we rule out such surgeries for a large class of Whitehead doubles. Furthermore, we rule out cosmetic surgeries for L-space knots along slopes with opposite signs.