# When do knots in light stay knotted?

## TL;DR

This paper establishes a mathematical condition under which initially knotted null light fields remain knotted over time, linking Maxwell's equations to fluid dynamics to guide the design of persistent knotted light fields.

## Contribution

It introduces a novel mapping between Maxwell's equations and fluid flow, providing a criterion for the persistence of knotted light fields and methods for their construction.

## Key findings

- Null light fields stay null if the initial Poynting flow is shear-free.
- A mapping between Maxwell's equations and Euler fluid flow is established.
- Conditions are provided for magnetic and electric field lines to be tangent to surfaces.

## Abstract

An initially knotted light field will stay knotted if it satisfies a set of nonlinear, geometric constraints, i.e. the null conditions, for all space-time. However, the question of when an initially null light field stays null has remained challenging to answer. By establishing a mapping between Maxwell's equations and transport along the flow of a pressureless Euler fluid, we show that an initially analytic null light field stays null if and only if the flow of the initial Poynting field is shear-free, giving a design rule for the construction of persistently knotted light fields. Furthermore we outline methods for constructing initially knotted null light fields, and initially null, shear-free light fields, and give sufficient conditions for the magnetic (or electric) field lines of a null light field to lie tangent to surfaces. Our results pave the way for the design of persistently knotted light fields and the study of their field line structure.