I may have some confusion about the twist and torsion of an elastic filament. The issue centers around this set-up.
I hold a cable in my hands, so that it forms a straight line. Keeping the endpoints clamped in my fingers, I move them closer together until they are right next to each other, so that the cable forms a circle. However, I can feel that there is significant twist-stress stored in the filament, so much so that for certain cables this is very hard to do. If I twist the endpoints as I move them together, though, I can feel that there is no stress, and the cable is clearly relaxed.
My question is, when is twist created in the first case? I have moved the endpoints in a straight line, and kept them clamped the whole time so they cannot rotate. It seems like the vanishing of the total twist should be topologically protected, but evidently not (or, the transformation I have performed on the filament somehow disturbed the topology). Where has my thinking gone wrong?
To be clear, my question is not about what, at the mechanical level, is causing the tension. I understand that as the material cross section rotates about the filament's tangent, stress is created. I'm more wondering how the cross section ends up rotating when I keep the endpoints clamped.