# Invariants of winding-numbers and steric obstruction in dynamics of flux lines

## TL;DR

This paper classifies the configuration sectors of 2D crossing flux lines, revealing that invariants and steric effects govern their dynamics, with invariants fully classifying configurations at low winding numbers.

## Contribution

It introduces a classification scheme based on polynomial invariants and steric effects for flux line configurations, advancing understanding of their dynamical obstructions.

## Key findings

- Invariants are polynomial functions of winding numbers.
- Steric obstruction prevents certain configuration transitions.
- Invariants fully classify configurations at low winding numbers.

## Abstract

We classify the sectors of configurations that result from the dynamics of 2d crossing flux lines, which are the simplest degrees of freedom of the 3-coloring lattice model. We show that the dynamical obstruction is the consequence of two effects: (i) conservation laws described by a set of invariants that are polynomials of the winding numbers of the loop configuration, (ii) steric obstruction that prevents paths between configurations, for lack of free space. We argue that the invariants fully classify the configurations in five, chiral and achiral, sectors and no further obstruction in the limit of low-winding numbers.