Higher Helicity of Magnetic Lines and Arf-invariants
Abstract
We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic -invariant. We present a simpler new proof (in part) that the -invariant is ergodic. The -invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a function of pairwise linking numbers of the magnetic lines. This property is based of the following fact: an arithmetic residue of the -invariant for a triple of closed magnetic lines (such a triple is a model of a link with even pairwise linking coefficients) coincides with the Arf-invariant. The new results concern magnetic fields on closed 3-dimensional manifolds and use the -invariant. To make this idea precise we generalize the Arf-invariants of classical semi-boundary links (including the Arf-Brown invariant) and we introduce a new Arf-invariant, called the hyperquaternionic Arf-invariant.
Keywords
Cite
@article{arxiv.1503.05306,
title = {Higher Helicity of Magnetic Lines and Arf-invariants},
author = {Petr M. Akhmetiev},
journal= {arXiv preprint arXiv:1503.05306},
year = {2015}
}