[math-fun] Curly loop puzzle

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? —Dan