Colorings of odd or even chirality on hexagonal lattices
  We define two classes of colorings that have odd or even chirality on hexagonal lattices. This parity is an invariant in the dynamics of all loops, and explains why standard Monte-Carlo algorithms are nonergodic. We argue that adding the motion of "stranded" loops allows for parity changes. By implementing this algorithm, we show that the even and odd classes have the same entropy. In general, they do not have the same number of states, except for the special geometry of long strips, where a Z$_2$ symmetry between even and odd states occurs in the thermodynamic limit. 
O. Cepas
cond-mat.stat-mech