By Alexander duality, $\mathbb{R}^3\setminus S$ has two connected components.

Let $u\colon \mathbb{R}^3\to \mathbb{S}^3\setminus (A\cup B)$ be the universal covering. Denote by $\tilde S$ a lift of $S$.

Again, by Alexander duality $\mathbb{R}^3\setminus \tilde S$ has two connected components. One of them, say $\Omega$ is bounded.

Observe that $u(\Omega)$ is one of the components of $\mathbb{R^3}\setminus S$ and it does not contain $A$ nor $B$.
It follows that both $A$ and $B$ lie in the other connected component — a contradiction.