On the characterising slopes of hyperbolic knots

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.

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Received 22 August 2018

Accepted 20 December 2018

Published 27 November 2019