  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.
