I. INTRODUCTION The q -state Potts model [1, 2] is a very important model in the study of phase transitions and critical phenomena. The critical frontier, or the loci of critical points, of the Potts model was first determined by Potts [1] for the square lattice. The critical exponents of the Potts model are obtained by conjectures on the basis of numerical evidence and by using Coulomb gas theory [3–7]. According to the universality hypothesis [8], the Potts model on different lattices belongs to the same universality class. But the determination of the critical frontier of the q -state Potts model in general, which includes the q = 1 bond and site percolation, is still an outstanding challenge. Particularly, the threshold of site percolation has remained largely unresolved. In a recent paper [9], hereafter referred to as I, one of us (FYW) considered the Potts model on two classes of very general two-dimensional lattices, the triangular-type and kagome-type lattices shown in Fig. 1. The Boltzmann weights W of the hatched trian- gles denote interactions involving 3 spins τ$_{1}$, τ$_{2}$, τ$_{3}$ = 1 , 2 , . . . , q surrounding a triangle, and are given by where δ$_{ij}$ = δ$_{Kr}$ ( s$_{i}$, s$_{j}$ ) , δ$_{123}$ = δ$_{12}$ δ$_{23}$ δ$_{31}$ , and A, B, C, A $^{′}$, B $^{′}$, C ′ are constants. Spin interac- tions within the hatched areas can be either 2- or 3-site couplings. The hatched triangles can have internal structures such as the stack-of-triangle subnets, which are of recent in- terest [10, 11], shown in Fig. 2. We refer to these structures as subnet networks . These stack-of-triangle lattices are called subnet lattices . Examples of triangular subnet lattices and kagome-type subnet lattices are shown in Fig. 2 and 3 of I. The 1 × 1 subnet lattices are the triangular and kagome lattices themselves. We shall call a kagome-type lattice with m × m down-pointing and n × n up-pointing subnets an ( m × m ) : ( n × n ) subnet lattice. In I, Wu derived closed-form expressions for the critical frontier of the q -state Potts model for the 2 types of lattices in Fig. 1. For the triangular-type lattices the critical frontier is exact, but for the kagome-type lattices the critical frontier is obtained under a homogeneity assumption. The purpose of this paper is two-fold: 2 W$_{△}$ (1 , 2 , 3) = A + B ( δ$_{12}$ + δ$_{23}$ + δ$_{31}$ ) + Cδ$_{123}$ , W$_{▽}$ (1 , 2 , 3) = A ′ + B $^{′}$( δ$_{12}$ + δ$_{23}$ + δ$_{31}$ ) + C $^{′}$δ$_{123}$ , (1)