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Handlebody complements in the 3-sphere: A remark on a theorem of Fox. (English) Zbl 0759.57012
The theorem of Fox mentioned in the title says that any connected 3- dimensional submanifold $$W$$ of the 3-sphere $$S^ 3$$ is homeomorphic to the complement of a union of handlebodies in $$S^ 3$$. Given additionally a collection $$C$$ of simple closed curves on the boundary of $$W$$, in the present note necessary and sufficient conditions (one extrinsic, one intrinsic) are given for $$W$$ to have an embedding in $$S^ 3$$ such that the complement $$S^ 3-W$$ is a union of handlebodies, and moreover $$C$$ contains a complete collection of meridians for these handlebodies. Note that, in case $$W$$ is the complement of a non-trivial knot in $$S^ 3$$, only one curve $$C$$ has this property by the recent solution of the knot complement conjecture, showing the difficulty of the problem in general.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M40 Characterizations of the Euclidean $$3$$-space and the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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