Totally tangential $\mathbb{C}$-links and electromagnetic knots

TL;DR

This paper introduces an algorithm to construct real-analytic Legendrian representatives for any link type, enabling explicit generation of electromagnetic knots and analyzing their dynamics within Bateman electromagnetic fields.

Contribution

It provides a systematic method to find Legendrian representatives for all link types and demonstrates how to derive electromagnetic field configurations from them.

Findings

Explicit algorithm for Legendrian link parametrization.

Linear system approach to construct electromagnetic fields.

Electromagnetic knots cannot be confined indefinitely in bounded regions.

Abstract

The set of real-analytic Legendrian links with respect to the standard contact structure on the 3-sphere $S^3$ corresponds both to the set of totally tangential $\mathbb{C}$-links as defined by Rudolph and to the set of stable knotted field lines in Bateman electromagnetic fields of Hopf type. It is known that every isotopy class has a real-analytic Legendrian representative, so that every link type $L$ admits a holomorphic function $G:\mathbb{C}^2\to\mathbb{C}$ whose zeros intersect $S^3$ tangentially in $L$ and there is a Bateman electromagnetic field $\mathbf{F}$ with closed field lines in the shape of $L$. However, so far the family of torus links are the only examples where explicit expressions of $G$ and $\mathbf{F}$ have been found. In this paper, we present an algorithm that finds for every given link type $L$ a real-analytic Legendrian representative, parametrised in terms of trigonometric polynomials. We then prove that (good candidates for) examples of $G$ and $\mathbf{F}$ can be obtained by solving a system of linear equations, which is homogeneous in the case of $G$ and inhomogeneous in the case of $\mathbf{F}$. We also use the real-analytic Legendrian parametrisations to study the dynamics of knots in Bateman electromagnetic fields of Hopf type. In particular, we show that no compact subset of $\mathbb{R}^3$ can contain an electromagnetic knot indefinitely.