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.


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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