There's an old result in knot theory that has a nice proof due to Conway (which I learned about from a Martin Gardner column ages ago): You can't tie nontrivial knots in different parts of a rope and pass them through each other in such a fashion that they cancel. (Can anyone provide a reference to the relevant column?) Proof idea: Create a tube that follows one knot and swallows the other. It occurs to me that this proof does not apply to configurations like the fisherman's knot (see the top picture in https://en.wikipedia.org/wiki/File:Fisherman's_knot.png). Or at least, it's not obvious to me how to apply Conway's trick in this context. (Maybe use a tube that isn't cylindrical, but bifurcates like a pair of pants? I don't see how to make this work.) So now I'm wondering: Is it possible to have something nontrivial like the fisherman's knot in one stretch of a two-rope cable, and something nontrivial in another stretch of the two-rope cable, and cancel them by appropriate manipulations of the cable? Jim Propp