Volume 5, issue 1 (2005)

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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
 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