Algebraic & Geometric Topology

Finite Dehn surgeries on knots in $S^3$

Yi Ni and Xingru Zhang

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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 1 .

Article information

Algebr. Geom. Topol., Volume 18, Number 1 (2018), 441-492.

Received: 22 November 2016
Revised: 20 June 2017
Accepted: 14 September 2017
First available in Project Euclid: 1 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

finite Dehn surgery Culler-Shalen norm Heegaard Floer homology


Ni, Yi; Zhang, Xingru. Finite Dehn surgeries on knots in $S^3$. Algebr. Geom. Topol. 18 (2018), no. 1, 441--492. doi:10.2140/agt.2018.18.441.

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