Bose–Mesner algebras attached to invertible Jones pairs

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways—by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to the so-called four-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on 4n vertices from each four-weight spin model on n vertices. s paper, we build on recent work with Munemasa to give a different proof to Yamadaʹs result, and we analyze the structure of the association scheme attached to this spin model.