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We classify the sectors of configurations that result from the dynamics of 2dcrossing flux lines, which are the simplest degrees of freedom of the3-coloring lattice model. We show that the dynamical obstruction is theconsequence of two effects: (i) conservation laws described by a set ofinvariants that are polynomials of the winding numbers of the loopconfiguration, (ii) steric obstruction that prevents paths betweenconfigurations, for lack of free space. We argue that the invariants fullyclassify the configurations in five, chiral and achiral, sectors and no furtherobstruction in the limit of low-winding numbers.

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