  We consider universal statistical properties of systems that are
characterized by phase states with macroscopic degeneracy of the ground state.
A possible topological order in such systems is described by non-linear
discrete equations. We focus on the discrete equations which take place in the
case of generalized exclusion principle statistics. We show that their exact
solutions are quantum dimensions of the irreducible representations of certain
quantum group. These solutions provide an example of the point where the
generalized exclusion principle statistics and braid statistics meet each
other. We propose a procedure to construct the quantum dimer models by means of
projection of the knotted field configurations that involved braiding features
of one-dimensional topology.
