We show that on a hyperbolic knot in , the distance between any two finite surgery slopes is at most , and consequently, there are at most three nontrivial finite surgeries. Moreover, in the case where admits three nontrivial finite surgeries, must be the pretzel knot . In the case where admits two noncyclic finite surgeries or two finite surgeries at distance , the two surgery slopes must be one of ten or seventeen specific pairs, respectively. For –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 and are characterizing slopes for the torus knot for each .
"Finite Dehn surgeries on knots in $S^3$." Algebr. Geom. Topol. 18 (1) 441 - 492, 2018. https://doi.org/10.2140/agt.2018.18.441