A note on band surgery and the signature of a knot

TL;DR

This paper investigates the effects of band surgery on knots, establishing a signature difference constraint for quasi-alternating knots with square-free determinants, and explores implications for chirally cosmetic bandings.

Contribution

It proves a new signature difference restriction for band surgeries on certain knots and links this to Heegaard Floer invariants, providing insights into chirally cosmetic bandings.

Findings

Signature difference is 0 or 8 for band surgeries on quasi-alternating knots with square-free determinant.

T(2,5) is the only square-free torus knot admitting a chirally cosmetic banding.

Chirally cosmetic bandings are rare and constrained by the established invariants.

Abstract

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer $d$-invariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that $T(2, 5)$ is the only torus knot $T(2, m)$ with $m$ square-free that admits a chirally cosmetic banding, i.e. a band surgery operation to its mirror image. We conclude with a discussion on the scarcity of chirally cosmetic bandings.