This is a more sensible (IMHO) restatement of this question:
Which Compact $3$-manifolds with boundaries embed in $\mathbb{S}^3?$ Is there any hope of a characterization?
This is a more sensible (IMHO) restatement of this question:
Which Compact $3$-manifolds with boundaries embed in $\mathbb{S}^3?$ Is there any hope of a characterization?
There is a theorem of Fox that more or less deals with this. Any such manifold is a complement of (possibly knotted) handlebodies.
Theorem: Every compact connected 3-submanifold $Y$ of the 3-sphere can be reimbedded in the 3-sphere so that the exterior of the image of $Y$ is a union of handlebodies, i.e. regular neighborhoods of embedded graphs.
R. H. Fox, On the imbedding of polyhedra in 3-space, Ann. of Math. (2) 49 (1948), 462–470.