A torus is a surface shaped like a donut. Imagine that I've painted two rings on a torus. One is on the outer surface, and goes through the hole in the center, coming around from the other side. The other ring is on the inner surface, and goes all the way around the hole in the center. These two rings of paint are linked.

I then cut a small hole in the torus. Through this hole, I turn the torus inside-out.

In the process, the rings of paint switch from the outer surface to the inner surface and vice versa. Therefore, they have become unlinked. How?

I'm afraid the first comment to this problem is correct.  This problem is in fact insoluble.  No topological process can unlink the two rings in 3 dimensions without cutting into the rings themselves.
So, unless you're cutting the hole such that it intersects one of the two rings, you won't unlink them.

Posted by AvalonXQ on 2007-01-07 01:41:30