From: Dan Asimov <asimov@msri.org> Let a C^oo closed curve in R^3 be called a "curly loop" if its curvature is nowhere vanishing. Two curly loops are for instance A) the unit circle in the plane and B) the unit circle in the plane traversed twice around. PUZZLE: Can A and B be continuously deformed one into the other ? through curly loops ? so that at all stages of the deformation the tangent directions vary continuously?
--Call a "noncurly loop" a C^oo closed curve in R^3 whose curvature is allowed to vanish. If you can do a curly-->curly deformation using noncurly loops, doesn't it automatically follow that you can do it with curly loops? I mean, just replace any tiny segment that looks locally like the graph y=x^3 or y=x^4 i.e has a zero-curvature point at x=0, with a helix-like vine wound round the old curve near there. In particular for your problem, your curve B could be "unfolded" 180 into a figure-8, then "untwisted" 180 into "an 8 with the middle sawn thru," then "inflated" into a circle A, where actually I'll assume slight distortions at all stages so that the curve never self-intersects and stays C^oo. That process may involve a small number of zero-curvature points on the curve, arising near the self intersection of the 8. But if in that vicinity we replace with a very highly wound helical patch during all times when there was any worry, then aren't we obviously safe?