Conformal geometry of curves and surfaces and geometric knot theory

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A one-parameter family of renormalized potentials of a compact domain of an Euclidean space is introduced. A couple of sufficient conditions for the uniqueness of a point that gives the maximum(or minimum) value of the potential are given. The renormalization of the average of the squares of the linking numbers of a given knot and random circles, which can be considered as generalization of the renormalization of the integration of a potential on the domain, is studied. It turns out to be invariant under Mobius transformations.