The Third Order Helicity of Magnetic Fields via Link Maps

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 `(m)]<sub>123</sub>{\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 `(m)]₁₂₃{\bar{\mu}_{123}} -invariant of orbits of B.