The third order helicity of magnetic fields via link maps

TL;DR

This paper introduces a new approach to third order helicity of magnetic fields using link invariants, providing bounds on energy and an ergodic interpretation based on link maps and configuration space invariants.

Contribution

It develops a novel geometric framework connecting third order helicity with Milnor invariants and homotopy invariants, offering new bounds and interpretations.

Findings

Provides a lower bound for the $L^2$-energy of magnetic fields.

Establishes a connection between helicity and Milnor's link invariants.

Offers an ergodic interpretation of third order helicity.

Abstract

We introduce an alternative approach to the third order helicity of a volume preserving vector field $B$, which leads us to a lower bound for the $L^2$-energy of $B$. The proposed approach exploits correspondence between the Milnor $\bar{\mu}_{123}$-invariant for 3-component links and the homotopy invariants of maps to configuration spaces, and we provide a simple geometric proof of this fact in the case of Borromean links. Based on these connections we develop a formulation for the third order helicity of $B$ on invariant \emph{unlinked} domains of $B$, and provide Arnold's style ergodic interpretation of this invariant as an average asymptotic $\bar{\mu}_{123}$-invariant of orbits of $B$.