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