If you untangle a thick, rather elastic rope in space, you may end up seeing the effects of torsional twists on it. It is as if the rope itself has been twisted, making it no longer straight. This would not be the case if it were untangled only in the surface on which it lies and never really lifted off the ground. In real life, such untangling would probably be done on the spur of the moment and might make the tangle worse. We might have given up before the end, because larger tangles are much harder than they first appear. Nevertheless, mathematics can model it in a simple, elegant and imaginative way that brings beautiful and interesting results. We show, that the maximal number of basic moves, required for passing between any two regularly homotopic planar or spherical curves with at most $n$ crossings, can be restricted within two quadratic bounds - functions of n. The explicit algorithm, which gives quadratic upper bound, chooses the path, along which all curves have at most n + 2 crossings. Furthermore, it offers a characterization of curves with minimal number of crossings in their homotopy class as well as a simple constructive proof of Whitney's theorem.
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