Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices

We study the critical frontiers of the Potts model on two-dimensional bow-tie lattices with fully inhomogeneous coupling constants. Generally, for the Potts critical frontier to be found exactly, the underlying lattice must be a three-uniform hypergraph. A more general class of lattices are the four-uniform ones, with unit cells contained within four boundary vertices. We demonstrate that in some cases, such lattices can be decomposed into triangular cells, and solved using a modification of standard techniques. This leads to the exact inhomogeneous Potts critical frontiers on various lattices, such as the bow-tie lattice with five different couplings, and critical points for asymmetric bow-tie lattices.