Three for the price of two?

Take a solid sphere. Its surface (or "skin") is homeomorphic to the sphere. Now drill a hole through the solid sphere:

The surface (or "skin") of the resulting solid is homeomorphic to the torus.

This time, drill two holes in the solid sphere, like this:

Now the surface (or "skin") of the resulting solid is homeomorphic to the 2-fold torus.

Question

Drill two holes in the solid sphere which "meet", like this:

To what is the surface (or "skin") of the resulting solid homeomorphic?

Notes

  1. The torus and 2-fold torus were easy to recognise as the earlier answers. Most people cannot just "spot" the answer in this more complicated question. So some topological technique is needed.
  2. You may assume the surface is orientable and has no boundary components.

Answer

Drawing further pictures made a big mess! So I replaced the original solid sphere with a solid cuboid (topologists won't object!). This makes pictures easier to draw, and also allowed me to actually make the objects and include photos and videos.

The solid in question looks like this:

Click the picture above for a video (no sound) of the solid being rotated. (Warning: it's an .avi file, ~ 8MB)

Now slice the block into two equal "halves", like in this video (again, .avi, ~7.5MB). Each of the resulting "halves" looks like this:

A bit of thought shows that the surface (or "skin") of each half has a subdivision:

Careful counting reveals that, for this subdivision, V=16, E=30, F=14. However, when the "halves" are reassembled (ie glued together to give the solid), twenty vertices are identified in pairs, ten edges are identified in pairs, and four faces are "destroyed". The effected parts are indicated in this picture of one of the halves:

Hence the surface (or "skin") of the required solid has a subdivision with

Accordingly, the Euler characteristic is 22-50+24=-4.

Given that the surface is orientable with no boundary componenets, it follows that its genus is (2--4)/2=3, and hence it is a 3-fold torus.