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arXiv · 0906.3494

The third order helicity of magnetic fields via link maps II

R. Komendarczyk
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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 $\barμ_{123}$-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 $\barμ_{123}$-invariant for parametrized Borromean links in $\R^3$.

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R. Komendarczyk. 2010-10-26. The third order helicity of magnetic fields via link maps II. https://doi.org/10.1063/1.3516611

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