Finite surgeries on three-tangle pretzel knots

Abstract

We classify Dehn surgeries on .p; q; r / pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit nontrivial finite surgeries are . 2; 3; 7/ and . 2; 3; 9/. Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p; q; r . We treat these using the Culler-Shalen norm of the SL.2; C/-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.