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We define two classes of colorings that have odd or even chirality onhexagonal lattices. This parity is an invariant in the dynamics of all loops,and explains why standard Monte-Carlo algorithms are nonergodic. We argue thatadding the motion of "stranded" loops allows for parity changes. Byimplementing this algorithm, we show that the even and odd classes have thesame entropy. In general, they do not have the same number of states, exceptfor the special geometry of long strips, where a Z$_2$ symmetry between evenand odd states occurs in the thermodynamic limit.

Colorings of odd or even chirality on hexagonal lattices