Are Maxwell knots integrable?

TL;DR

This paper explores the properties of knotted field lines in source-free Maxwell equations, proposing that their evolution might be integrable due to conserved invariants like the Abelian link invariant.

Contribution

It demonstrates that the Poynting vector induces an integrable evolution of knotted electromagnetic field lines and suggests the existence of multiple conserved invariants.

Findings

The Abelian link invariant is an integral of motion for the field lines.

Ranada's solution with Hopf links is likely exceptional among knotted solutions.

Most field lines are not closed and do not form knots or links, except for a measure-zero set.

Abstract

We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. The same is expected to be true also for the non-Abelian invariants (like Jones and HOMFLY-PT polynomials or Vassiliev invariants), and many integrals of motion can imply that the Poynting evolution is actually integrable. We also consider particular examples of the field lines for the particular family of finite energy source-free "knot" solutions, attempting to understand when the field lines are closed -- and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada's solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist.