Extremal polygons

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Date(s) - 17 janvier 2013

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The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ in many cases is not a perfect Morse function. Surptisingly, for 3D configurations the situation becomes nicer: for an equilateral linkage with odd number of edges the area function is always a perfect Morse function and fits the lacunary principle. Therefore cyclic equilateral polygons can be interpreted as independent generators of the homology groups of the (decorated) configuration space.