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How would I calculate the topology of a sphere, with a smaller sphere inside removed.

I know if I drill through to get to the hole, then we are back to being a sphere, and if I drill out to the other side (i.e. all the way through) then we get torus. We if I drilled to the hollowed out inner sphere and dug two different tunnels out, then what is the genus?

I don't really know how to draw this better but here is my attempt at drawing profil slice of it.

https://docs.google.com/drawings/d/1WjGjZQWq7tFhpqebZnVwqMDAPCrXNjJC_Jy-2KabzO8/edit

If you held the sphere and looked down a hole u would just see a sphere with a hole drilled all the way through, but all the holes meet though in the in the middle.

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Once you've drilled the first hole, you have a solid torus. If you drill through it a second time, you effectively drill two holes through the torus. In the end, you've added two new holes to your object, and its surface has genus three. –  Olivier Bégassat May 17 '13 at 20:26
    
If you want to make your comment and answer, I can choose it as correct. –  Steven-Owen May 17 '13 at 23:37
    
@OlivierBégassat: I take it you're assuming the tunnels don't intersect? –  HSN May 18 '13 at 1:12

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