Non-integer characterizing slopes for torus knots

A slope $p/q$ is a characterizing 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 for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K^\prime$ in $S^3$ have homeomorphic $p/q$-surgeries, then for $q \geq 3$ and $p$ sufficiently large we can conclude that $K$ and $K^\prime$ have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.

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Received 9 October 2017

Accepted 2 April 2018

Published 7 December 2020