Characterizing slopes for satellite knots

TL;DR

This paper investigates when Dehn surgeries on satellite knots uniquely determine the knot, showing that for large enough slopes, especially non-integral ones, the surgery's outcome characterizes the knot.

Contribution

It extends previous results to satellite knots, proving that all sufficiently large slopes are characterizing, with a focus on non-integral slopes for composite knots.

Findings

Any slope with sufficiently large |q| is characterizing for a knot.

Every non-integral slope is characterizing for a composite knot.

The approach uses JSJ decomposition and constraints on surgery slopes.

Abstract

A slope $p/q$ is said to be characterizing for a knot $K$ if the homeomorphism type of the $p/q$-Dehn surgery along $K$ determines the knot up to isotopy. Extending previous work of Lackenby and McCoy on hyperbolic and torus knots respectively, we study satellite knots to show that for a knot $K$, any slope $p/q$ is characterizing provided $|q|$ is sufficiently large. In particular, we establish that every non-integral slope is characterizing for a composite knot. Our approach consists of a detailed examination of the JSJ decomposition of a surgery along a knot, combined with results from other authors giving constraints on surgery slopes that yield manifolds containing certain surfaces.