Narrow Results

An L operator is presented related to an infinite dimensional limit of the fusion R matrices for and . It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated.

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (N_α,N_β) model. This model consists of two scalar Potts models (of N_α and N_β states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If N_α,N_β≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for N_α=N_β; whereas if N_α≠N_β=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

It is shown that the Jones polynomial occurring in the theory of links as a knot invariant is expressible directly as the partition function of a vertex model and a q²-state Potts model. This formulation of the Jones polynomial, which makes use of the solution of the Yang-Baxter equation of a q-state vertex model considered previously by Perk and Wu, does not require the use of the writhe of the link as needed in previous statistical mechanical considerations.

Four approaches to construct polynomial invariants for trivalent knotted graphs in S³ are compared. The first approach is based on vertex model with R-matrices and Glebsh-Gordan coefficients, appearing in SL_q(2)-representations theory, as Boltzman weights. The second approach is based on Kauffman's quantum spin network theory, the third one is based on Witten-Turaev area-coloring model (or face model) based on quantum 6j-symbols, where q is root of unity. The fourth approach is based on the same face (or area-coloring) model, but q is not root of unity. The coincidence (up to certain normalization) of topological invariants, arising from these four state models, is proved.

Using the vertex model interpretation of the coloured (generalised) Jones polynomial of a link L, we show that if the colour of the i^th component is N_i+m_ir, then modulo t^r−1 this coloured Jones polynomial is congruent, up to a product of calculable factors, to the coloured Jones polynomial with the colour of the i^th component N_i, where N_i and r are positive integers, and m_i is a non-negative integer.

The proof depends on the fact that, up to a known factor, the coloured Jones polynomial of a link may be calculated from a (1, 1)-tangle, the closure of which represents the link.

The phase diagram and the critical indices are investigated for the symmetric 16-vertex model on the square lattice by combining a variational series expansion and the coherent anomaly method. The nonuniversal critical exponents smoothly interpolate between two exactly solvable cases, namely the Baxter eight-vertex model and the Ising model.