Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B ⊆ G; we call B a basis of G. We introduce a two-parameter graph polynomial P B (q, v) that depends on B and its embedding in G. The algebraic curve P B (q, v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = e K − 1, defined on G. This curve predicts the phase diagram not only in the physical ferromagnetic regime (v > 0), but also in the antiferromagnetic (v < 0) region, where analytical results are often difficult to obtain. For larger bases B the approximations become increasingly accurate, and we conjecture that P B (q, v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G-or for the Ising model (q = 2) on any G-the polynomial P B (q, v) factorises for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P B (q, v) can be used to detect exact solvability of the Potts model on G.We illustrate the method for two choices of G: the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker-Kadanoff phase at certain Beraha numbers. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wu's approach. We perform large-scale numerical computations for comparison and find excellent agreement with the polynomial predictions. For v > 0 the accuracy of the predicted critical coupling v c is of the order 10 −4 or 10 −5 for the 6-edge basis, and improves to 10 −6 or 10 −7 for the largest basis studied (with 36 edges).We dedicate this article to Professor Fa-Yueh Wu on the occasion of his 80th birthday.

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3, 122 ) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagomé and (3, 122 ) bond problems, and the other gives estimates of pc for the homogeneous kagomé (0.5244088...) and (3,12 2 ) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.

Abstract.In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial P B (q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic twodimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = e K − 1 of P B (q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, P B (q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P B (q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures v c obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain v c (4, 82 ) = 3.742 489 (4), v c (kagome) = 1.876 459 7 (2), and v c (3, 122 ) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.

Percolation thresholds have recently been studied by means of a graph polynomial P B (p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P B (p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P B (p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P B (p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion.We present bond percolation polynomials for the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p c (4, 8 2 ) = 0.676 803 329 · · ·, p c (kagome) = 0.524 404 998 · · ·, p c (3, 12 2 ) = 0.740 420 798 · · ·, comparable to the best simulation results. We also show that the alternative definition of P B (p) can be applied to study site percolation problems.

Abstract. Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.