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