Finite Dehn surgeries on knots in S3

Ni, Yi; Zhang, Xingru

doi:10.2140/agt.2018.18.441

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Finite Dehn surgeries on knots in $S^3$

Yi Ni and Xingru Zhang

Algebraic & Geometric Topology 18 (2018) 441–492

DOI: 10.2140/agt.2018.18.441

Abstract

We show that on a hyperbolic knot $K$ in $S^{3}$ , 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 $4 m$ and $4 m + 4$ are characterizing slopes for the torus knot $T (2 m + 1, 2)$ for each $m \geq 1$ .

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Keywords

finite Dehn surgery, Culler-Shalen norm, Heegaard Floer homology

Mathematical Subject Classification 2010

Primary: 57M25

References

Publication

Received: 22 November 2016

Revised: 20 June 2017

Accepted: 14 September 2017

Published: 10 January 2018

Authors

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

Volume 18, issue 1 (2018)