#### 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
##### 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 $P\left(-2,3,7\right)$. 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 $T\left(2m+1,2\right)$ for each $m\ge 1$.

##### Keywords
finite Dehn surgery, Culler-Shalen norm, Heegaard Floer homology
Primary: 57M25