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.

[1]  On Four-Weight Spin Models and their Gauge Transformations , 2000 .

[2]  Eiichi Bannai,et al.  Generalized generalized spin models (four-weight spin models) , 1995 .

[3]  Vaughan F. R. Jones,et al.  On knot invariants related to some statistical mechanical models , 1989 .

[4]  V. Jones.  A polynomial invariant for knots via von Neumann algebras , 1985 .

[5]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[6]  Kazumasa Nomura,et al.  An Algebra Associated with a Spin Model , 1997 .

[7]  François Jaeger,et al.  Strongly regular graphs and spin models for the Kauffman polynomial , 1992 .

[8]  Eiichi Bannai,et al.  Spin Models on Finite Cyclic Groups , 1994 .

[9]  François Jaeger,et al.  New constructions of models for link invariants , 1996 .

[10]  Pierre de la Harpe,et al.  Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model. , 1994 .

[11]  A. Neumaier,et al.  Distance Regular Graphs , 1989 .

[12]  François Jaeger,et al.  On Spin Models, Triply Regular Association Schemes, and Duality , 1995 .

[13]  F. Jaeger.  Towards a Classification of Spin Models in terms of Association Schemes , 1996 .

[14]  Kazumasa Nomura,et al.  Bose-Mesner Algebras Related to Type II Matrices and Spin Models , 1998 .

[15]  Eiichi Bannai,et al.  On Spin Models, Modular Invariance, and Duality , 1997 .

[16]  K. Nomura,et al.  Spin Models Constructed from Hadamard Matrices , 1994, J. Comb. Theory, Ser. A.