Purely cosmetic surgeries and Casson--Walker--Lescop invariants

TL;DR

This paper investigates the limitations on purely cosmetic surgeries on null-homologous knots in rational homology spheres using the Casson--Walker--Lescop invariant, providing new constraints and classifications.

Contribution

It introduces new bounds on the number of purely cosmetic surgeries for null-homologous knots in rational homology spheres utilizing the Casson--Walker--Lescop invariant.

Findings

Any null-homologous knot admits at most two pairs of purely cosmetic surgeries.

Constraints are established for knots in manifolds with first Betti number one or two.

In certain manifolds, at most two inequivalent knots share the same exterior.

Abstract

Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the $3$-sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We also present constraints for null-homologous knots in certain $3$-manifolds with the first Betti number one or two to admit purely cosmetic surgeries. As another application, we show that, for a null-homologous knot in some $3$-manifolds, including $S^2 \times S^1$, there are at most two knots which are inequivalent to the given one, but whose exteriors are orientation-preservingly homeomorphic to that of the given one.