Some Formulas for Spin Models on Distance-Regular Graphs

A spin model is a square matrixWsatisfying 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 modelWof an association scheme which containsWin its Bose?Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose?Mesner algebraN(W). In this paper we study the caseW?A?N(W), where A is the Bose?Mesner algebra of a distance-regular graph. We show the following results. Let?=(X,R) be a distance-regular graph of diameterd>1 such that the Bose?Mesner algebra A of?satisfiesW?A?N(W) for some spin modelWonX. WriteW=?di=0tiAi, whereAidenotes theith adjacency matrix. Setxi=t?1i?1tiandp=x?11x2. Thenxi=pi?1x1holds for alli. Moreover, the eigenvalues and the intersection numbers of?are rational functions ofx1andp.

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