Tying knots in light fields

TL;DR

This paper introduces a new class of null solutions to Maxwell's equations in free space that encode all torus knots and links, demonstrating topology-preserving evolution and complex geometric structures.

Contribution

It combines Bateman and spinor formalisms with complex polynomials on  to construct and analyze knotted electromagnetic field solutions with preserved topology.

Findings

Constructed null solutions encoding all torus knots and links.

Showed that the evolution preserves the topology of the knots and links.

Illustrated the geometric structure of nested knotted tori filled by the field lines.

Abstract

We construct a new family of null solutions to Maxwell's equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is both geodesic and shear-free, preserves the topology of the knots and links. Our approach combines the Bateman and spinor formalisms for the construction of null fields with complex polynomials on $\mathbb{S}^3$. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.