Finite dehn surgeries on knots in S³

We show that on a hyperbolic knot K in S³, 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 4m and 4m+4 are characterizing slopes for the torus knot T (2m+1,2) for each m ≥ 1.