Cosmetic surgery in L–space homology spheres

Abstract

Let $K$ be a nontrivial knot in $S^3$, and let $r$ and $r^{\prime }$ be two distinct rational numbers of same sign. We prove that there is no orientation-preserving homeomorphism between the manifolds $S_r^3\left(K\right)$ and $S_{r^{\prime }}^3\left(K\right)$. We further generalize this uniqueness result to knots in arbitrary L–space homology spheres.