## Algebraic & Geometric Topology

### Finite Dehn surgeries on knots in $S^3$

#### 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 $m ≥ 1$.

#### Article information

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

Dates
Revised: 20 June 2017
Accepted: 14 September 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.agt/1517454222

Digital Object Identifier
doi:10.2140/agt.2018.18.441

Mathematical Reviews number (MathSciNet)
MR3748249

Zentralblatt MATH identifier
06828010

#### Citation

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. https://projecteuclid.org/euclid.agt/1517454222

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