I am reading a paper called "JSJ-decomposition of knot and link complements in $S^3$", written by Ryan Budney. My question does not concern the essence of the paper but a technical fact about 3-manifolds at the very beginning of the paper. The statement of a theorem starts as follows :
Let $M$ be a connected compact sub-manifold of $S^3$ with $\partial M$ a disjoint union of $n$ tori. By Alexander's Theorem, $\overline{S^3 \setminus M}$ consists of a disjoint union of $p$ solid tori and $q$ non-trivial knot complements, where $p+q=n$.
This is followed by a conclusion for the theorem. I do not immediately see why the complement of $M$ decomposes as such. The only theorem of Alexander I know that concerns tori in the $3$-sphere is the one stating that any torus $S^1 \times S^1 \subset S^3$ always bounds a solid torus $S^1 \times D^2$.
Please tell me if you find a reason for this, or if there is another Theorem of Alexander to use here.
Thanks in advance !