Some Formulas for Spin Models on Distance-Regular Graphs

✍ Scribed by Brian Curtin; Kazumasa Nomura

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 213 KB
• Volume: 75
• Category: Article
• ISSN: 0095-8956
• DOI: 10.1006/jctb.1998.1876

✦ Synopsis

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= d i=0 t i A i , where A i denotes the ith adjacency matrix. Set x i =t &1 i&1 t i and p=x &1 1 x 2 . Then x i = p i&1 x 1 holds for all i. Moreover, the eigenvalues and the intersection numbers of 1 are rational functions of x 1 and p.