A Dehn surgery is called a Seifert fibering surgery if it yields a Seifert fibered manifold. It has been conjectured that nontrivial, Seifert fibering sugeries on knots in the 3-sphere are integral surgeries unless the knot is a trivial knot, a torus knot, or a cable of a torus knot. We first prove an analogous result for knots in a solid torus. As a corollary it is shown that the conjecture holds if a regular or exceptional fiber of the resulting Seifert fibered manifold is unknotted in the (original) 3-sphere; this assumption is verified for many Seifert fibering surgeries. As another application, we show that except for trivial examples, no periodic knots with period greater than 2 produce a Seifert fibered manifold with an infinite fundamental group by surgery.
Introduction.Let K be a knot in a 3-manifold M. A slope of K is the isotopy class of a simple closed curve on dN(K). The manifold obtained from M by Dehn surgery on a knot K with slope 7 is denoted by M ( In this paper we first prove a result analogous to Conjecture 1.1 for knots in a solid torus. A 0-bridge braid in a solid torus V is an essential simple closed curve isotopic to a curve in the boundary of V. A core of V is a 0-bridge braid, so that any other 0-bridge braid is a cable of a 0-bridge braid.
Theorem 1.2. Let K be a knot in a solid torus V such that K is not contained in a 3-ball in V. Suppose that V(K; 7) is a Seifert fibered manifold where the slope 7 is not meridional. Then one of the following holds.(1) K is a core ofV or a cable of a 0-bridge braid in V.(2) 7 is an integral slope.
Abstract. We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (−3, 3, 5)-pretzel knot belongs to both of the infinite families.
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