All integral slopes can be Seifert fibered slopes for hyperbolic knots

Motegi, Kimihiko; Song, Hyun-Jong

doi:10.2140/agt.2005.5.369

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All integral slopes can be Seifert fibered slopes for hyperbolic knots

Kimihiko Motegi and Hyun-Jong Song

Algebraic & Geometric Topology 5 (2005) 369–378

DOI: 10.2140/agt.2005.5.369

arXiv: math.GT/0505322

Abstract

Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the $3$ –sphere $S^{3}$ ? It is conjectured that if $r$ –surgery on a hyperbolic knot in $S^{3}$ yields a Seifert fiber space, then $r$ is an integer. We show that for each integer $n \in ℤ$ , there exists a tunnel number one, hyperbolic knot $K_{n}$ in $S^{3}$ such that $n$ –surgery on $K_{n}$ produces a small Seifert fiber space.

Keywords

Dehn surgery, hyperbolic knot, Seifert fiber space, surgery slopes

Mathematical Subject Classification 2000

Primary: 57M25, 57M50

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Publication

Received: 10 March 2005

Revised: 25 March 2005

Accepted: 12 April 2005

Published: 30 April 2005

Authors

Kimihiko Motegi
Department of Mathematics Nihon University Tokyo 156-8550 Japan

Hyun-Jong Song
Division of Mathematical Sciences Pukyong National University 599-1 Daeyondong Namgu Pusan 608-737 Korea

Volume 5, issue 1 (2005)