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If you take a sphere and drill a hole through it, the shape can be continuously deformed to a torus of genus = 1. A sphere with two separate holes is homeomorphic to a torus with genus 2.

I am wondering about a similar kind of shape where you drill three holes into a sphere that are connected to one another. Is there a deformation to a shape that is more recognizable to a non-mathematician?

user18764
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    The great thing is that, even if you are having trouble seeing how to deform in this into a "standard" genus g torus, you can compute the genus just by making a polygonal version and computing $F+V-E = 2-2g$. I will not ruin the fun for you. Maybe once you know what the answer should be, you will be able to see the homeomorphism. – Steven Gubkin Jan 25 '14 at 04:45

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First of all, I think that you may be confusing "sphere" with "ball". If you take a sphere a drill a hole through it, you get a cylinder, not a torus. But if you take a ball and drill a hole through it, you get a $3$-manifold with boundary whose boundary is a torus.

If you take a ball and drill two holes through it intersecting at the center of the sphere, you get a $3$-manifold whose boundary is a genus $3$ surface. To see this, imagine stretching one of the holes very much, so that what you have is essentially a donut with a hole drilled through it perpendicularly to its axis of rotational symmetry. There are three holes: one in the center, and two on opposite sides of the donut.

Doing the same thing with three holes intersecting at the center will leave you with two extra holes, so in other words with a genus $5$ surface.

Bruno Joyal
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