Finite surgeries on three-tangle pretzel knots

TL;DR

This paper classifies specific Dehn surgeries on pretzel knots that produce manifolds with finite fundamental groups, identifying only two hyperbolic pretzel knots with such surgeries and developing new techniques involving character varieties.

Contribution

It introduces novel methods for detecting boundary slopes via the SL(2,C)-character variety and completes the classification of finite surgeries on three-tangle pretzel knots.

Findings

Only (-2,3,7) and (-2,3,9) pretzel knots admit non-trivial finite surgeries.

New techniques for boundary slope detection using the Culler-Shalen norm.

Reduction of the problem to small indices p,q,r using the 6-theorem.

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 non-trivial 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.