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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

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, )–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.

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