Tangent-point self-avoidance energies for curves

We study a two-point self-avoidance energy Eq which is defined for all rectifiable curves in Rn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of Eq(γ) for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle S1 or to a closed interval I. For q> 2 the energy Eq evaluated on curves in R3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in R3 with finite Eq-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent qcrit = 2, the arclength parametrization of γ is of class C1,1−2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1−2/q is optimal