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 $P\left(-2,3,7\right)$. 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 $m\ge 1$.