All integral slopes can be Seifert fibered slopes for hyperbolic knots. (English) Zbl 1083.57012

For a hyperbolic knot in the \(3\)-sphere, all but finitely many slopes yield hyperbolic manifolds by Thurston’s hyperbolic Dehn surgery theorem. Exceptional surgeries are split into four cases, according to the resulting manifold: a reducible manifold, a Seifert fiber space, a toroidal manifold or a counterexample to the geometrization conjecture. The first and last ones are expected not to happen. Also, it is conjectured that a Seifert fiber space is only obtained by an integral slope.
The paper under review shows that for each integer \(n\), there exists a tunnel number one hyperbolic knot on which \(n\)-surgery yields a Seifert fiber space over the \(2\)-sphere with three exceptional fibers. The analogous result for toroidal surgery was already known by M. Teragaito [Proc. Am. Math. Soc. 130, No. 9, 2803–2808 (2002; Zbl 0997.57033)]. The proof is constructive, and uses the Montesinos trick to show that the resulting manifold is a Seifert fiber space. Finally, it is verified that there exists a trivial knot disjoint from the knot which becomes a Seifert fiber in the resulting Seifert fiber space. This supports a conjecture by K. Miyazaki and K. Motegi [Commun. Anal. Geom. 7, No. 3, 551–583 (1999; Zbl 0940.57025)].


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds


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