#### Volume 9, issue 2 (2009)

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Finite surgeries on three-tangle pretzel knots

### David Futer, Masaharu Ishikawa, Yuichi Kabaya, Thomas W Mattman and Koya Shimokawa

Algebraic & Geometric Topology 9 (2009) 743–771
##### 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.

 Dedicated to Professor Akio Kawauchi on the occasion of his 60th birthday.
##### Keywords
pretzel knot, exceptional Dehn surgery, finite surgery, Culler–Shalen seminorm
##### Mathematical Subject Classification 2000
Primary: 57M05, 57M25, 57M50
##### Publication
Received: 29 September 2008
Accepted: 10 February 2009
Published: 20 April 2009
##### Authors
 David Futer Mathematics Department Temple University Philadelphia, PA 19122 USA Masaharu Ishikawa Mathematical Institute Tohoku University Sendai, 980-8578 Japan Yuichi Kabaya Department of Mathematics Tokyo Institute of Technology 2-12-1 Oh-okayama, Meguro-ku Tokyo 152-8551 Japan Thomas W Mattman Department of Mathematics and Statistics California State University at Chico Chico, CA 95929-0525 USA Koya Shimokawa Department of Mathematics Saitama University Saitama 338-8570 Japan