  We study a constrained statistical-mechanical model in two dimensions that
has three useful descriptions. They are 1) the Ising model on the honeycomb
lattice, constrained to have three up spins and three down spins on every
hexagon, 2) the three-color/fully-packed-loop model on the links of the
honeycomb lattice, with loops around a single hexagon forbidden, and 3) three
Ising models on interleaved triangular lattices, with domain walls of the
different Ising models not allowed to cross. Unlike the three-color model, the
configuration space on the sphere or plane is connected under local moves. On
higher-genus surfaces there are infinitely many dynamical sectors, labeled by a
noncontractible set of nonintersecting loops. We demonstrate that at infinite
temperature the transfer matrix admits an unusual structure related to a gauge
symmetry for the same model on an anisotropic lattice. This enables us to
diagonalize the original transfer matrix for up to 36 sites, finding an entropy
per plaquette ~ 0.3661 and substantial evidence that the model is not critical.
We also find the striking property that the eigenvalues of the transfer matrix
on an anisotropic lattice are given in terms of Fibonacci numbers. We comment
on the possibility of a topological phase, with infinite topological
degeneracy, in an associated two-dimensional quantum model.
