Handlebody complements in the 3-sphere: a remark on a theorem of Fox

Scharlemann, Martin

doi:10.1090/S0002-9939-1992-1116272-X

Handlebody complements in the $3$-sphere: a remark on a theorem of Fox
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by Martin Scharlemann

Proc. Amer. Math. Soc. 115 (1992), 1115-1117

DOI: https://doi.org/10.1090/S0002-9939-1992-1116272-X

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Abstract:

Let $W$ be a compact $3$-dimensional submanifold of ${S^3}$, and $C$ be a collection of disjoint simple closed curves on $\partial W$. We give necessary and sufficient conditions (one extrinsic, one intrinsic) for $W$ to have an imbedding in ${S^3}$ so that ${S^3} - W$ is a union of handlebodies, and $C$ contains a complete collection of meridia for these handlebodies.

References

Ralph H. Fox, On the imbedding of polyhedra in $3$-space, Ann. of Math. (2) 49 (1948), 462–470. MR 26326, DOI 10.2307/1969291
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7