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 ${\mathbb{R}}^3$, 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 ${\mathbb{R}}^3$.

19 pages, 2 figures; published version

References in corpus (4)

• The third order helicity of magnetic fields via link maps
• Asymptotic Vassiliev Invariants for Vector Fields
• Lie coalgebras and rational homotopy theory II: Hopf invariants
• Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

Cited by in corpus (5)

• Pontryagin invariants and integral formulas for Milnor's triple linking number
• On volume-preserving vector fields and finite type invariants of knots
• Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows, Part 2
• The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids
• Homotopy string links and the $\kappa$-invariant