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The Third Order Helicity of Magnetic Fields via Link Maps

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, Univ. of Pennsylvania, Philadelphia, PA, 19104, USA

    R. Komendarczyk

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  1. R. Komendarczyk
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Correspondence to R. Komendarczyk.

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Communicated by P.T. Chruściel

This project is supported by DARPA, #FA9550-08-1-0386.

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Komendarczyk, R. The Third Order Helicity of Magnetic Fields via Link Maps. Commun. Math. Phys. 292, 431–456 (2009). https://doi.org/10.1007/s00220-009-0896-z

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