Combinatorial formula for the $M$ invariant of magnetic lines

TL;DR

This paper introduces a combinatorial formula for the $M$ invariant of magnetic lines, crucial for MHD problems, by leveraging invariants of classical links without relying on analytic integrals.

Contribution

It presents a novel combinatorial approach to compute the $M$ invariant, avoiding complex integrals and focusing on asymptotic properties of classical link invariants.

Findings

The formula is verified through calculations on simple examples.

The approach confirms the asymptotic property without using analytic integrals.

The method simplifies the computation of the $M$ invariant in magnetic line studies.

Abstract

To solve MHD problems within the framework of the theory of two-scale mean fields, it is important to study the invariants of magnetic lines. Such invariants are constructed on the basis of invariants of classical links, which must satisfy the asymptotic property. We choose the simplest asymptotic invariant $M_3$ of three-component links, which is not expressed in terms of the pairwise linking coefficients of the components. We check the asymptotic property based only on the combinatorial definition of the invariant and do not use the analytic integral. For simple examples, the proven formula is verified by calculation.