Finite surgeries on three-tangle pretzel knots

Abstract

We classify Dehn surgeries on $\left(p,q,r\right)$ pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit nontrivial finite surgeries are $\left(-2,3,7\right)$ and $\left(-2,3,9\right)$. 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 $SL\left(2,ℂ\right)$–character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.