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The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1-skeleton of a triangulation of an open disk. G is said to be CP parabolic respectively CP hyperbolic], if there is a locally nite disk packing P in the plane respectively, the unit disk] with contacts graph G. Several criteria(More)
Let P be a locally finite disk pattern on the complex plane C whose com-binatorics is described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function Θ : E → [0, π/2] on the set of edges. Let P * be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function.(More)
A physically natural potential energy for simple closed curves in R 3 is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist(More)