Volume 18, issue 1 (2018)

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Finite Dehn surgeries on knots in $S^3$

Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 18 (2018) 441–492

We show that on a hyperbolic knot K in S3, the distance between any two finite surgery slopes is at most 2, and consequently, there are at most three nontrivial finite surgeries. Moreover, in the case where K admits three nontrivial finite surgeries, K  must be the pretzel knot P(2,3,7). In the case where K admits two noncyclic finite surgeries or two finite surgeries at distance 2, the two surgery slopes must be one of ten or seventeen specific pairs, respectively. For D–type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that 4m and 4m + 4 are characterizing slopes for the torus knot T(2m + 1,2) for each m 1.

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finite Dehn surgery, Culler-Shalen norm, Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 57M25
Received: 22 November 2016
Revised: 20 June 2017
Accepted: 14 September 2017
Published: 10 January 2018
Yi Ni
Department of Mathematics
California Institute of Technology
Pasadena, CA
United States
Xingru Zhang
Department of Mathematics
University at Buffalo
Buffalo, NY
United States