Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E (α) , α ∈ [2, 3)

We develop a precise analysis of J. O’Hara’s knot functionals E (α) , α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E (α) on injective and regular H 2 curves and then we establish Fréchet differentiability of E (α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E (2) , the so‐called Möbius Energy, we prove C ∞ ‐smoothness of critical points of the appropriately rescaled functionals \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$\end{document} by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.