Tuesday, May 26, 2009

The unlinking torus

A little known fact: If you have a torus (a donut-shaped tube) with a hole in it, then it is possible to turn it inside-out. I will leave the details for you to work out, but rest assured that it is possible.

That's the inspiration for this puzzle (which I basically stole from Martin Gardner).

Here we have a torus. There's a big hole in it so that we can turn it inside-out. There is also a ring (in red) painted on the outside of the torus. There is also a ring (in dark blue) painted on the inside of the torus. These two rings are linked.

But suppose I turned the tube inside-out. Then the red ring will be on the inside, while the blue ring will be on the outside. Then the rings will no longer be linked!

Did I really manage to unlink the rings? If so, how?

See the solution

8 comments:

The Barefoot Bum said...

Your condition that the rings are "painted on" the torus seems ambiguous; you might want to qualify this condition more precisely.

miller said...

The paint forms an unbroken ring. It is attached to the surface of the torus; the red ring to the outside surface, the blue ring to the inside surface. Ask for any more clarifications if needed.

intrinsicallyknotted said...

As I've seen this puzzle before (in Martin Gardner's book!), I will refrain from giving it away. But to clarify the answer to Barefoot Bum's question, assume that the torus actually has some thickness, like an inner tube made of rubber, while the rings are truly one-dimensional. Both lie on the surface of the solid object that is the inner tube. Then since the rings are separated by at least the thickness of the rubber at all points, they are unambiguously non-intersecting.

Baumann Eduard said...

I too have the Gardner book "!aha aha!". I'm looking forward for a beautyfull series of drawings when Miller presents the solution.

Jeffrey Ellis said...

My head hurts. Damn you, Miller!

Scott said...

I too love Gardner, although I haven't seen this particular puzzle. After some doodling, I think the answer is no: they remain linked. But I won't spoil the puzzle by saying why...

Secret Squïrrel said...

I don't know much about topology and less about knots (intrinsic or otherwise), but I am positive that if two rings are linked, there is no way to unlink them by manipulating them in 3-space.

I tried imagining that the hole was as large as possible so that I could see what happens to the torus when it is everted but I had difficulty picturing the complete surface. Nonetheless, I could see that the two rings (all that remained after I enlarged the hole!) were still linked.

I'll await your answer to make things clearer (as you always do).

Scott said...

Hi Squirrel -- you and I thought alike about this. Rather than trying to mentally "evert" the surface, try just thinking about different ways to re-close the hole.