Cable knots do not admit cosmetic surgeries

TL;DR

This paper proves that cable knots in the 3-sphere do not admit purely cosmetic Dehn surgeries, confirming Gordon's conjecture for this class of knots.

Contribution

It establishes that non-trivial cable knots cannot have purely cosmetic surgeries, advancing the understanding of Dehn surgery uniqueness.

Findings

Cable knots do not admit purely cosmetic surgeries

Confirms Gordon's conjecture for cable knots

Supports the uniqueness of Dehn surgeries on non-trivial knots

Abstract

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.