On Spin Models, Triply Regular Association Schemes, and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using seriesparallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.