Finite surgeries on three-tangle pretzel knots

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.

Dedicated to Professor Akio Kawauchi on the occasion of his 60th birthday.

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Keywords

pretzel knot, exceptional Dehn surgery, finite surgery, Culler–Shalen seminorm

Mathematical Subject Classification 2000

Primary: 57M05, 57M25, 57M50

References

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

Volume 9, issue 2 (2009)

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

Abstract