Strongly regular graphs and spin models for the Kauffman polynomial (Q1205426)

The author obtains specializations of the Kauffman polynomial for links, depending on the choice of a formally self-dual strongly regular graph with strongly regular subconstituents, as studied by \textit{P. J. Cameron}, \textit{J. M. Goethals} and \textit{J. J. Seidel} [J. Algebra 55, 257-280 (1978; Zbl 0444.05045)]. These graphs are rather scarce, and most of the known ones lead to known specializations of the Kauffman polynomial. The Higman-Sims graph however gives a new example. The connection between the graphs and the link invariants is obtained via spin models. The adjacency matrix of the given graph generates a 3- dimensional algebra which is closed under the componentwise Hadamard product, a so-called Bose-Meisner algebra, with suitable properties. Those properties make it possible to re-interpret the algebra as coming from a different type of generating matrix, which may be combinatorially interpreted as a spin model. This in turn yields a direct combinatorial definition of a link invariant. In order to establish this connection and, in particular, to single out the right classes of Bose-Meisner algebras and of graphs, a systematic study of the relationship between spin models and association schemes is given.