On the characterising slopes of hyperbolic knots

TL;DR

This paper demonstrates that most slopes used in Dehn surgeries on hyperbolic knots are characterising, meaning they uniquely determine the knot, especially highlighting stronger results for hyperbolic L-space knots.

Contribution

It extends the understanding of characterising slopes for hyperbolic knots, showing that all but finitely many slopes are characterising, with stronger results for hyperbolic L-space knots.

Findings

Most slopes with denominator q ≥ 3 are characterising for hyperbolic knots.

All but finitely many non-integer slopes are characterising for hyperbolic L-space knots.

Combines Lackenby's results with Heegaard Floer homology to establish these properties.

Abstract

A slope $p/q$ is a characterising slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that when $K$ is a hyperbolic knot its set of characterising slopes contains all but finitely many slopes $p/q$ with $q \geq 3$. We prove stronger results for hyperbolic $L$-space knots, showing that all but finitely many non-integer slopes are characterising. The proof is obtained by combining Lackenby's proof that for a hyperbolic knot any slope $p/q$ with $q$ sufficiently large is characterising with genus bounds derived from Heegaard Floer homology.