  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
3-uniform hypergraph. A more general class of lattices are the 4-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.
