A note on band surgery and the signature of a knot

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 $K^{\prime }$ 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.

The main theorem has been strengthened. This version accepted for publication in the Bulletin of the London Mathematical Society