I would like to define a notion of a topological tunnel, but I don't know how (or even if it is possible) to capture it topologically. I am interested in closed 2-manifolds in $\mathbb{R^3}$. Suppose you have a solid block of material, and you drill a hole in it. The hole is like a tube or tunnel, it enters at one spot, exits at another, and in between can take any path (even knotted) that does not self-intersect or touch the surface until the tunnel exits. Now drill another such tunnel, same rules, but now it cannot intersect or touch the previous tunnel, but it may weave around it. Etc. The shape of the tunnel is irrelevant, but I want it to be independent of others.
I don't think genus captures this notion of a tunnel. For example, holes shaped like the letter 'Y', or the letter 'H', can never occur with my tunnels. Is there a concept used in topology that corresponds to these tunnels? If not, can you see how to unambiguously define a tunnel?
Thanks for any help!