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Characterizing slopes for torus knots

Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 14 (2014) 1249–1274

A slope p q is called a characterizing slope for a given knot K0 in S3 if whenever the p q–surgery on a knot K in S3 is homeomorphic to the p q–surgery on K0 via an orientation preserving homeomorphism, then K = K0. In this paper we try to find characterizing slopes for torus knots Tr,s. We show that any slope p q which is larger than the number 30(r2 1)(s2 1)67 is a characterizing slope for Tr,s. The proof uses Heegaard Floer homology and Agol–Lackenby’s 6–theorem. In the case of  T5,2, we obtain more specific information about its set of characterizing slopes by applying further Heegaard Floer homology techniques.

Dehn surgery, torus knots, characterizing slopes, Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 57M27, 57R58
Secondary: 57M50
Received: 1 December 2012
Revised: 22 July 2013
Accepted: 29 July 2013
Published: 7 April 2014
Yi Ni
Department of Mathematics
California Institute of Technology
1200 E California Blvd
Pasadena, CA 91125
Xingru Zhang
Department of Mathematics
University at Buffalo
Buffalo, NY 14260