Characterizing slopes for torus knots

Abstract

A slope $\frac{p}{q}$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac{p}{q}$–surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac{p}{q}$–surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In this paper we try to find characterizing slopes for torus knots $T_{r,s}$. We show that any slope $\frac{p}{q}$ which is larger than the number $30\left(r^2-1\right)\left(s^2-1\right)∕67$ is a characterizing slope for $T_{r,s}$. The proof uses Heegaard Floer homology and Agol–Lackenby’s $6$–theorem. In the case of $T_{5,2}$, we obtain more specific information about its set of characterizing slopes by applying further Heegaard Floer homology techniques.