Quantization of energy and writhe in self-repellingknots

Abstract

Probably the most natural energy functional to be considered forknotted strings is that given by electrostatic repulsion. Inthe absence of counter-charges, a charged, knotted stringevolving along the energy gradient of electrostatic repulsionwould progressively tighten its knotted domain into a point on aperfectly circular string. However, in the presence of chargescreening self-repelling knotted strings can be stabilized.It is known that energy functionals in which repulsive forcesbetween repelling charges grow inversely proportionally to thethird or higher power of their relative distance stabilizeself-repelling knots. Especially interesting is the case of thethird power since the repulsive energy becomes scale invariantand does not change upon Möbius transformations(reflections in spheres) of knotted trajectories. We observehere that knots minimizing their repulsive Möbius energyshow quantization of the energy and writhe (measure ofchirality) within several tested families of knots.