On Four-Weight Spin Models and their Gauge Transformations

We study the four-weight spin models (W1, W2, W3, W4) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W1, W3), or the same pair (W2, W4). As a consequence, we show that the link invariant associated with a four-weight spin model is not sensitive to the full reversal of orientation of a link. We also show in a similar way that such a link invariant is invariant under mutation of links. Next, we give an algebraic characterization of the transformations of four-weight spin models which preserve W1, W3 or preserve W2, W4. Such “gauge transformations” correspond to multiplication of W2, W4 by permutation matrices representing certain symmetries of the spin model, and to conjugation of W1, W3 by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W1, W3 are symmetric. Finally we apply these results to two-weight spin models obtained as solutions of the modular invariance equation for a given Bose-Mesner algebra B and a given duality of B. We show that the set of such spin models is invariant under certain gauge transformations associated with the permutation matrices in B. In the case where B is the Bose-Mesner algebra of some Abelian group association scheme, we also show that any two such spin models (which generalize those introduced by Eiichi and Etsuko Bannai in J. Alg. Combin. 3 (1994), 243–259) are related by a gauge transformation. As a consequence, the link invariant associated with such a spin model depends only trivially on the link orientation.