The topology of stable electromagnetic structures and Legendrian fields on the 3-sphere

TL;DR

This paper explores the topology of null solutions to Maxwell's equations on the 3-sphere, linking electromagnetic field lines with Legendrian fields and contact topology, and constructs solutions with knotted and toroidal structures.

Contribution

It establishes a connection between null electromagnetic solutions and Legendrian fields on the 3-sphere, characterizes when Legendrian links can be realized as field lines, and constructs solutions with knotted and toroidal structures.

Findings

Legendrian links with vanishing rotation number can be realized as electric or magnetic field lines.

Any Seifert foliation of S^3 is isotopic to a foliation by Legendrian fields.

Constructed null solutions with knotted electric lines and toroidal magnetic surfaces.

Abstract

Null solutions to Maxwell's equations in free space have the property that the topology of the electric and magnetic lines is preserved for all time. In this article we connect the study of a particularly relevant class of null solutions (related to the Hopf fibration) with the existence of pairs of volume preserving Legendrian fields with respect to the standard contact structure on the 3-sphere. Exploiting this connection, we prove that a Legendrian link can be realized as a set of closed orbits of a non-vanishing Legendrian field corresponding to the electric or magnetic part of a null solution if and only if each of its components has vanishing rotation number. Moreover, we prove that any foliation by circles (a Seifert foliation) of $S^3$ is isotopic to the foliation defined by a volume preserving Legendrian field with respect to the standard contact structure. We also construct a new null solution to the Maxwell's equations with the property that every positive torus knot is a closed electric line of its electric field. Finally, we prove that any (possibly knotted) toroidal surface in $\mathbb{R}^3$ can be realized as a magnetic surface of a null solution to Maxwell's equations, thus implying its stability for all times. In particular, the associated volume preserving Legendrian field on $S^3$ exhibits a positive volume set of invariant tori of the same topological type.