2015/03/18 by Petr M. Akhmetiev, Akhmetiev, Petr M.
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1503.05306
arxiv created 2015/03/30 · arxiv updated 2015/03/31
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.