Piecewise circular curves and Positivity

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We introduce the moduli space of generic circular n -gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when n = 2 k , this moduli space contains a connected component homeomorphic to the Fock–Goncharov space of k -tuples of positive flags for PSp ( 4 , ℝ ) and hence is a topological ball. We characterize this component geometrically as the space of simple circular n -gons with decreasing curvature.