The third order helicity of magnetic fields via link maps II

R. Komendarczyk

Published 2009-06-18, updated 2010-10-26Version 2

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 $\R^3$.