Annals of Combinatorics 4 (2000) 401-412

Some Remarks on Four-Weight Spin Models

Tayuan Huang and Haitao Guo

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan

Graduate School of Mathematics, Kyushu University, Fukuoka 812, Japan

Received March 20, 1998

AMS Subject Classification: 05E30

Abstract. 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, 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.

Keywords: four-weight spin models, Bose-Mesner Algebra, association schemes


1.  E. Bannai and T. Ito, Algebraic Combinatorics I, Association Schemes, Benjamin Cummings, Menlo Park, 1984.

2.  E. Bannai and E. Bannai, Generalized spin models and association schemes, Mem. Fac. Science Kyushu Univ. A 47 (2) (1993) 397–409.

3.  E. Bannai and E. Bannai, Generalized spin models (four-weight spin models), Pacific J. of Math. 170 (1995) 1–16.

4.  E. Bannai, E. Bannai, and F. Jaeger, On spin models, modular invariance, and duality, J. of Algebraic Combinatorics, 6 (1997) 203–228.

5.  A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, 1989.

6.  D.G. Higman, Coherent algebras, Linear Algebra Appl. 93 (1987) 209–239.

7.  F. Jaeger, Strongly regular graphs and spin models for the Kauffman polynomial, Geom. Dedicata, 44 (1992) 23–52.

8.  F. Jaeger, Spin models for link invariants,Lecture Notes Series Vol. 218, P. Rowlinson, Ed., London Mathematical Society, 1995, pp. 71–101.

9.  F. Jaeger, On spin models, triply regular association schemes, and duality, J. of Algebraic Combinatorics 4 (1995) 103–144.

10.  F. Jaeger, M. Matsumoto, and K. Nomura, Bose–Mesner algebras related to type II matrices and spin models, J. of Algebraic Combinatorics 8 (1998) 39–92.

11.  F. Jaeger, Toward a classification of spin models in terms of association schemes, in: Progress in Algebraic Combinatorics, Advanced Studies in Pure Mathematics Vol. 24, E. Bannai and A. Munemasa, Eds., Mathematical Society of Japan, 1996, pp. 197–225.

12.  F. Jaeger, On four-wieght spin models and their gauge-transformations, to appear.

13.  V. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math, 137 (1989) 311–334.

14.  K. Kawagoe, A. Munemasa, and Y. Watatani, Generalized spin models, J. of Knot Theory and its Ramifications 3 (1994) 465–475.

15.  K. Nomura, An algebra associated with a spin model, J. of Algebraic Combinatorics, 6 (1997) 53–88.

16.  P. Terwilliger, The subconstituent algebra of an association scheme, I, II, III, J. Algebraic Combinatorics 1 (1992) 363–388; 2 (1993) 73–103, 177–210.

Get the DVI | PS | PDF file of this abstract.