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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!

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Perhaps this is describable as a complement of the arcs in a tangle (suitably expanded to have a "width")? One problem is that the definition of tangle requires the solid block to be a 3-sphere. – Zev Chonoles May 15 '11 at 23:30
@Zev: I didn't think of defining it via the complement! Nice idea--Thanks! – Joseph O'Rourke May 15 '11 at 23:34
@Zev: so I believe what Joseph is looking for is a slightly larger notion. Pretty suitable terminology would be tangle in a handlebody. This is meant to describe an embedding of $\sqcup_k [0,1]$ in $H$ where $H$ is a 3-dimensional handlebody. You require that the embedding is proper in that the boundary of the disjoint union of the intervals is sent to the boundary of the handlebody, etc. The natural equivalence relation would be taking such embeddings up to a 1-parameter family (ambient isotopy rel boundary). – Ryan Budney May 16 '11 at 0:05
That said, a tangle in a handlebody is just a regular tangle in the 3-ball, but where a certain sub-tangle is the trivial tangle. – Ryan Budney May 16 '11 at 0:06
@Ryan & @Zev: This is wonderful! I think you collared the concept! Thanks so much! – Joseph O'Rourke May 16 '11 at 0:11

This question has been answered in comments. The relevant topological concept is that of a tangle (in a ball/handlebody/3-manifold with boundary, depending on the choice of the original solid block).

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