Transfer matrices for the partition function of the Potts model on cyclic and Möbius lattice strips

We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G,q,ν), for arbitrary q and temperature variable ν, on cyclic and Möbius strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width L_y vertices and of arbitrarily great length L_x vertices. For the cyclic case we express the partition function as Z(λ,L_y × L_x,q,ν) = ∑_d=0^Lyc^(d) Tr[(T _Z,λ,Ly,d)^m], where λ denotes lattice type, c^(d) are specified polynomials of degree d in q, T _Z,λ,Ly,d is the transfer matrix in the degree-d subspace, and m = L_x (L_x/2) for λ = sq, tri (hc), respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating T_Z,λ,Ly,d for arbitrary L_y. Explicit results for arbitrary L_y are given for T_Z,λ,Ly,d with d = L_y and L_y - 1. In particular, we find very simple formulas the determinant det(T_Z,λ,Ly,d), and trace Tr(T _Z,λ,Ly). Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included. We also present formulas for self-dual cyclic strips of the square lattice.