Quantization of energy and writhe in self-repelling knots

Abstract

Probably the most natural energy functional to be considered for knotted strings is that given by electrostatic repulsion. In the absence of counter-charges, a charged, knotted string evolving along the energy gradient of electrostatic repulsion would progressively tighten its knotted domain into a point on a perfectly circular string. However, in the presence of charge screening self-repelling knotted strings can be stabilized. It is known that energy functionals in which repulsive forces between repelling charges grow inversely proportionally to the third or higher power of their relative distance stabilize self-repelling knots. Especially interesting is the case of the third power since the repulsive energy becomes scale invariant and does not change upon Möbius transformations (reflections in spheres) of knotted trajectories. We observe here that knots minimizing their repulsive Möbius energy show quantization of the energy and writhe (measure of chirality) within several tested families of knots.