Symmetric Versus Non-Symmetric Spin Models for Link Invariants

We study spin models as introduced in [20]. Such a spin model can be defined as a square matrix satisfying certain equations, and can be used to compute an associated link invariant. The link invariant associated with a symmetric spin model depends only trivially on link orientation. This property also holds for quasi-symmetric spin models, which are obtained from symmetric spin models by certain gauge transformations preserving the associated link invariant. Using a recent result of [16] which asserts that every spin model belongs to some Bose-Mesner algebra with duality, we show that the transposition of a spin model can be realized by a permutation of rows. We call the order of this permutation the index of the spin model. We show that spin models of odd index are quasi-symmetric. Next, we give a general form for spin models of index 2 which implies that they are associated with a certain class of symmetric spin models. The symmetric Hadamard spin models of [21] belong to this class and this leads to the introduction of non-symmetric Hadamard spin models. These spin models give the first known example where the associated link invariant depends non-trivially on link orientation. We show that a non-symmetric Hadamard spin model belongs to a certain triply regular Bose-Mesner algebra of dimension 5 with duality, and we use this to give an explicit formula for the associated link invariant involving the Jones polynomial.