On the characterising slopes of hyperbolic knots

A slope $p\mathrm{/}q$ is a characterising slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p\mathrm{/}q$-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 $p\mathrm{/}q$ with $q\ge 3$. 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 $p\mathrm{/}q$ with $q$ sufficiently large is characterising with genus bounds derived from Heegaard Floer homology.