Properties of null knotted solutions to Maxwell's equations

Abstract

We discuss null knotted solutions to Maxwell's equations, their creation through Bateman's construction, and their relation to the Hopf-fibration. These solutions have well-known, conserved properties, related to their winding numbers. For example: energy; momentum; angular momentum; and helicity. The current research has focused on Lipkin's zilches, a set of little-known, conserved quantities within electromagnetic theory that has been explored mathematically, but over which there is still considerable debate regarding physical interpretation. The aim of this work is to contribute to the discussion of these knotted solutions of Maxwell's equations by examining the relation between the knots, the zilches, and their symmetries through Noether's theorem. We show that the zilches demonstrate either linear or more complicated relations to the p-q winding numbers of torus knots, and can be written in terms of the total energy of the electromagnetic field. As part of this work, a systematic multipole expansion of the vector potential of the knotted solutions is being carried out.