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An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this video.

I have heard some testimonial evidence that it is also possible to turn your pants backwards, again with the constraint of having your feet tied together. This second claim seems pretty dubious to me.

Question. Is it indeed possible to turn your pants backwards, while having your feet tied together by a piece of string? A set of instructions or a video demonstration would suffice for a yes answer. A precise mathematical formulation of the problem together with a proof of impossibility would suffice for a no answer.

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To avoid seeing something I might rather not: is that video American or British? – Loop Space Apr 19 '10 at 7:11
It's perfectly decent. – Elizabeth S. Q. Goodman Apr 19 '10 at 7:49
up vote 25 down vote accepted

I think that the answer is no, by consideration of linking numbers. First simplify the human body plus cord joining the ankles to a circle, and assign it an orientation. Also assign an orientation to each pant cuff. This can be done, e.g., so that each cuff has linking number +1 with the "body" (in which case the two cuffs are oppositely oriented).

Now suppose that there is an isotopy of the pants that turns them backwards. This means the left cuff is now on the right ankle and vice versa. But this also reverses the linking numbers, which is impossible.

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Thanks a lot John. The argument looks right to me. – Tony Huynh Apr 19 '10 at 3:56
You're welcome, Tony. Thanks for asking such an irresistible question! – John Stillwell Apr 19 '10 at 4:22
Neat, that simplification (with the cuffs connected at a point or line segment) also gets me to see why the inside-out procedure works. Of course I'm tempted to say "rotate the cuffs around the circle" (ha), but in fact you mostly just have to pass one cuff through the other at the bottom of the circle. – Elizabeth S. Q. Goodman Apr 19 '10 at 7:48

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