Some Formulas for Spin Models on Distance-Regular Graphs

A spin model is a square matrix W satisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin model W of an association scheme which contains W in its Bose Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose Mesner algebra N(W). In this paper we study the case W # A N(W), where A is the Bose Mesner algebra of a distance-regular graph. We show the following results. Let 1=(X, R) be a distance-regular graph of diameter d>1 such that the Bose Mesner algebra A of 1 satisfies W # A N(W) for some spin model W on X. Write W= i=0 tiAi , where Ai denotes the ith adjacency matrix. Set xi=t i&1 ti and p=x &1 1 x2 . Then xi= p x1 holds for all i. Moreover, the eigenvalues and the intersection numbers of 1 are rational functions of x1 and p. 1999 Academic Press