Characterizing slopes for torus knots

Ni, Yi; Zhang, Xingru

doi:10.2140/agt.2014.14.1249

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Characterizing slopes for torus knots

Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 14 (2014) 1249–1274

DOI: 10.2140/agt.2014.14.1249

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 $30 (r^{2} - 1) (s^{2} - 1) ∕ 67$ 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.

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)