Repulsive knot energies and pseudodifferential calculus for O'Hara's knot energy family \(E^{(\alpha )}, \alpha \in [2, 3)\) (Q2888905)

Knot energy functionals were originally introduced to simplify knots experimentally. Further aims regarding energies were in studying critical knots (perfect forms) for energy functionals and appropriate gradient and descent flows taking an arbitrary knot to a perfect form. This paper is dedicated to smoothness and variational aspects of J. O'Hara's knot functionals \(E^{(\alpha)}\), \(\alpha\in [2, 3)\). The author derives continuity of \(E^{(\alpha)}\) on injective and regular \(H^2\) knots, establishes Fréchet differentiability of \(E^{(\alpha)}\) and then states several first variation formulae. He proves \(C^\infty\)-smoothness of critical points of the appropriately rescaled functionals equation image by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.