Four-weight Spin Models Research Articles

Characterization of SDP Designs That Yield Certain Spin Models

We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of the cardinalities of intersections of four blocks.

On quasi-3 designs and spin models

A theorem of Bannai and Sawano shows that certain four-weight spin models exist if and only if certain quasi-3 designs exist. We verify that the known quasi-3 designs, other than SDP designs, do not give rise to four-weight spin models.

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 4 n vertices from each four-weight spin model on n vertices. In this paper, we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyze the structure of the association scheme attached to this spin model.

Four-weight spin models and Jones pairs

We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a “dual” pair of association schemes.

Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W_1, W_2, W_3, W_4;D) with the property that the entries of the matrix W_2 (or equivalently W_4) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W_2 if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W_2.

Bose-Mesner Algebras Associated with Four-Weight Spin Models

It is known that for each matrix W i and it's transpose t W i in any four-weight spin model (X, W 1, W 2, W 3, W 4; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W i ) and N( t W i ) = N′(W i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W 1) = N′(W 1) = N(W 3) = N′(W 3) holds. In this paper we show that all of them coincide, that is, N(W i ), N′(W i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra.

Some classes of four-weight spin models

Abstract We classify four-weight spin models (X;W1,W2,W3,W4) with sizes at most four. We also study the class of four-weight spin models with exactly two distinct values in W4, reduce their existence to that of certain symmetric designs.

Some Remarks on Four-Weight Spin Models

As a generalization of symmetric spin models, a four-weight spin model (X; W1, W2, W3, W4) consists of a finite set X and four non-zero complex matrices indexed by elements of X satisfying certain conditions set for polynomial invariants of links and knots in R3 through their partition functions. We show that W4tW4 = tW4W4, W1°tW1 lie in a certain symmetric Bose-Mesner algebra using spectral techniques. Based on this algebra, we then show that entries of W1 were essentially determined by its subconstituent algebra.

The Classification of Certain Four-Weight Spin Models

In this paper we show that if one of the matrices {Wi, 1 ≤i≤ 4} of a four-weight spin model (X, W1, W2, W3, W4; D) is equivalent to the matrix of a Potts model or a cyclic model as type II matrix and |X|≥ 5, then the spin model is gauge equivalent to a Potts model or a cyclic model up to simultaneous permutations on rows and columns. Using this fact and Nomura's result [12] we show that every four-weight spin model of size |X|= 5 is gauge equivalent to either a Potts model or a cyclic model up to simultaneous permutations on rows and columns.

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.

Generalized generalized spin models (four-weight spin models)

Generalized generalized spin models (four-weight spin models)

Twisted Extensions of Spin Models

A spin model is one of the statistical mechanical models which were introduced by V.F.R. Jones to construct invariants of links. In this paper, we give a new construction of spin models of size 4n from a given spin model of size n. The process is similar to taking tensor product with a spin model of size four, but we add some sign exchange. This construction also gives symmetric four-weight spin models of the type introduced by E. Bannai and E. Bannai.