#### Volume 14, issue 3 (2014)

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Characterizing slopes for torus knots

### Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 14 (2014) 1249–1274
##### 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.

##### Keywords
Dehn surgery, torus knots, characterizing slopes, Heegaard Floer homology
##### Mathematical Subject Classification 2010
Primary: 57M27, 57R58
Secondary: 57M50