Characterizing slopes for torus knots

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 $30 (r^{2} - 1) (s^{2} - 1) ∕ 67$ 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.

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Keywords

Dehn surgery, torus knots, characterizing slopes, Heegaard Floer homology

Mathematical Subject Classification 2010

Primary: 57M27, 57R58

Secondary: 57M50

References

Publication

Received: 1 December 2012

Revised: 22 July 2013

Accepted: 29 July 2013

Published: 7 April 2014

Authors

Yi Ni
Department of Mathematics California Institute of Technology 1200 E California Blvd Pasadena, CA 91125 USA
http://www.its.caltech.edu/~yini/
Xingru Zhang
Department of Mathematics University at Buffalo Buffalo, NY 14260 USA
http://www.math.buffalo.edu/~xinzhang/

Volume 14, issue 3 (2014)

Yi Ni and Xingru Zhang

Abstract