Cosmetic Surgery in Integral Homology $L$-Spaces

TL;DR

This paper proves that for a non-trivial knot in the 3-sphere, the manifolds obtained by Dehn surgery with different rational slopes are not homeomorphic, extending the result to knots in integral homology L-spaces.

Contribution

It establishes a uniqueness result for Dehn surgeries on knots in integral homology L-spaces, generalizing previous results for the 3-sphere.

Findings

No orientation-preserving homeomorphism exists between manifolds from different rational surgeries on the same knot.

The result applies to knots in arbitrary integral homology L-spaces.

The proof extends known surgery uniqueness results beyond the 3-sphere context.

Abstract

Let $K$ be a non-trivial knot in $S^3$, and let $r$ and $r'$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; we prove that there is no orientation-preserving homeomorphism between the manifolds $S^3_r(K)$ and $S^3_{r'}(K)$. We further generalize this uniqueness result to knots in arbitrary integral homology L-spaces.