The third order helicity of magnetic fields via link maps II

TL;DR

This paper extends the third order helicity concept for magnetic fields supported on unlinked domains, providing practical formulas, an ergodic interpretation, and energy bounds, with applications to Milnor invariants and Borromean links.

Contribution

It introduces a new formula for third order helicity using de Rham cohomology, applicable to unlinked domains, and connects it to ergodic theory and energy bounds.

Findings

Derived a practical formula for third order helicity.

Connected helicity to ergodic average of Milnor invariants.

Established an $L^2$-energy bound for magnetic fields.

Abstract

In this sequel we extend the derivation of the third order helicity to magnetic fields supported on unlinked domains in 3-space. The formula is expressed in terms of generators of the deRham cohomology of the configuration space of three points in $\R^3$, which is a more practical domain from the perspective of applications. It also admits an ergodic interpretation as an average asymptotic Milnor $\bar{\mu}_{123}$-invariant and allows us to obtain the $L^2$-energy bound for the magnetic field. As an intermediate step we derive an integral formula for Milnor $\bar{\mu}_{123}$-invariant for parametrized Borromean links in $\R^3$.