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Small knots in Seifert fibered 3-manifolds. (English) Zbl 0789.57009

Constructing knots with given properties in arbitrary 3-manifolds turns out to be a rather difficult problem. The question we consider here is: at which conditions does an arbitrary 3-manifold contain a small knot, i.e. a knot without closed incompressible surfaces in its exterior excepted peripheral tori? Actually one can hope that such knots exist at least in non-Haken 3-manifolds; this may be interesting in this case since such knots have a property which is very close to the one possessed by the ambient manifold.
In this paper we prove that non-trivial small knots exist in non-Haken Seifert fibered 3-manifolds (by non-trivial knots we mean knots that are not fibers of the Seifert fibration).
Reviewer: L.M.Lopez (Tokyo)

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:

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