Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup\_i H^1\_i) \cup \{\text{2-handles}\}$ such that:


1. $L \cap H^0$ is a set of trivial arcs;
2. $L \cap H^1\_i$ is either empty or the core of $H^1\_i$ for all $i$;
3. $L$ does not intersect the 2-handles?



Handlebody decomposition of a 3-manifold adapted to a link
Take a Heegaard splitting of the link exterior and then glue in solid tori with your link components as cores. The Heegaard splitting survives as a Heegaard splitting of the filled manifold and since the link is disjoint from the Heegaard surface it is disjoint from the 2-handles. To see that the other properties hold, you just have to consider the compressionbody obtained by Dehn filling torus boundary components of a compressionbody.


