We show how to compute the exact partition function for lattice
statistical-mechanical models whose Boltzmann weights obey a special "crossing"
symmetry. The crossing symmetry equates partition functions on different
trivalent graphs, allowing a transformation to a graph where the partition
function is easily computed. The simplest example is counting the number of
nets without ends on the honeycomb lattice, including a weight per branching.
Other examples include an Ising model on the Kagome' lattice with three-spin
interactions, dimers on any graph of corner-sharing triangles, and non-crossing
loops on the honeycomb lattice, where multiple loops on each edge are allowed.
We give several methods for obtaining models with this crossing symmetry, one
utilizing discrete groups and another anyon fusion rules. We also present
results indicating that for models which deviate slightly from having crossing
symmetry, a real-space decimation (renormalization-group-like) procedure
restores the crossing symmetry