Communications in Analysis and Geometry

Volume 28 (2020)

Number 7

Non-integer characterizing slopes for torus knots

Pages: 1647 – 1682



Duncan McCoy (Départment de Mathématiques, Université du Québec à Montréal, QC, Canada)


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.

Received 9 October 2017

Accepted 2 April 2018

Published 7 December 2020