The third order helicity of magnetic fields via link maps II

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 $\R^3$.