Higher Helicity of Magnetic Lines and Arf-invariants

TL;DR

This paper explores higher helicity invariants of magnetic fields, introduces a new Arf-invariant called the hyperquaternionic Arf-invariant, and demonstrates their properties and relationships on closed 3-manifolds.

Contribution

It presents a simpler proof of the ergodicity of the $M$-invariant and introduces a new hyperquaternionic Arf-invariant extending classical invariants.

Findings

The $M$-invariant is ergodic.

The $M$-invariant's residue relates to the Arf-invariant.

Introduction of the hyperquaternionic Arf-invariant.

Abstract

We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic $M$-invariant. We present a simpler new proof (in part) that the $M$-invariant is ergodic. The $M$-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 $M$-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 $M$-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.