Let $M$ be a compact 3-manifold and $L \subset M$ a link. A tunnel system for $L$ is a set $T = \{ t_1,...,t_n\}$ of embedded arcs in $M$ with endpoints on $L$ such that after removing an open regular neighborhood of $L \cup T$ from $M$ we obtain a handlebody. For $M = S^3$ if we take a projection of $L$ then we can just add tunnels at each of the crossings in order to obtain a tunnel system.
How do we prove that tunnel systems exist in general?