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

Abstract

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 H² 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 $\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$ by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.