Symmetric Versus Non-Symmetric Spin Models for Link Invariants
François Jaeger
and Kazumasa Nomura
DOI: 10.1023/A:1018771332556
Abstract
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.
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.Pages: 241–278
Keywords: spin model; link invariant; Bose-mesner algebra
Full Text: PDF
References
1. E. Bannai and Et. Bannai, “Generalized generalized spin models (four-weight spin models),” Pacific J. Math. 170 (1995), 1-16.
2. E. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994), 243-259.
3. E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Alg. Combin., 6 (1997), 203-228.
4. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
5. Et. Bannai and A. Munemasa, “Duality maps of finite abelian groups and their applications to spin models,” J. Alg. Combin., to appear.
6. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
7. T. Deguchi, “Generalized generalized spin models associated with exactly solvable models,” in Progress in Algebraic Combinatorics, E. Bannai and A. Munemasa (Eds.), Advanced Studies in Pure Math. Mathematical Society of Japan, 1996, Vol. 24, pp. 81-100.
8. G.V. Epifanov, “Reduction of a plane graph to an edge by a star-triangle transformation,” Soviet Math. Doklady 7 (1966), 13-17.
9. P. de la Harpe, “Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model,” Pacific J. Math. 162 (1994), 57-96.
10. F. Jaeger, “Strongly regular graphs and spin models for the Kauffman polynomial,” Geom. Dedicata 44 (1992), 23-52.
11. F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin. 4 (1995), 103-144.
12. F. Jaeger, “Spin models for link invariants,” in Surveys in Combinatorics 1995, P. Rowlinson (Ed.), London Mathematical Society Lecture Notes Series, Cambridge University Press, 1995, Vol. 218, pp. 71-101.
13. F. Jaeger, “Towards a classification of spin models in terms of association schemes,” in Progress in Algebraic Combinatorics, E. Bannai and A. Munemasa (Eds.), Advanced Studies in Pure Math. Mathematical Society of Japan, 1996, Vol. 24, pp. 197-225.
14. F. Jaeger, “New constructions of models for link invariants,” Pacific J. Math. 176 (1996), 71-116.
15. F. Jaeger, “On four-weight spin models and their gauge transformations,” J. Alg. Combin., to appear.
16. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Alg. Combin. 8 (1998), 39-72.
17. V.F.R. Jones, “A polynomial invariant for knots via von Neumann algebras,” Bull. Am. Math. Soc. 12 (1985), 103-111.
18. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pacific J. Math. 137 (1989), 311-336.
19. L.H. Kauffman, “State models and the Jones polynomial,” Topology 26 (1987), 395-407.
20. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications 3 (1994), 465-475.
21. K. Nomura, “Spin models constructed from Hadamard matrices,” J. Combin. Theory Ser. A 68 (1994), 251-261.
22. K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin. 6 (1997), 53-58.
2. E. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994), 243-259.
3. E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Alg. Combin., 6 (1997), 203-228.
4. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park, 1984.
5. Et. Bannai and A. Munemasa, “Duality maps of finite abelian groups and their applications to spin models,” J. Alg. Combin., to appear.
6. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
7. T. Deguchi, “Generalized generalized spin models associated with exactly solvable models,” in Progress in Algebraic Combinatorics, E. Bannai and A. Munemasa (Eds.), Advanced Studies in Pure Math. Mathematical Society of Japan, 1996, Vol. 24, pp. 81-100.
8. G.V. Epifanov, “Reduction of a plane graph to an edge by a star-triangle transformation,” Soviet Math. Doklady 7 (1966), 13-17.
9. P. de la Harpe, “Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model,” Pacific J. Math. 162 (1994), 57-96.
10. F. Jaeger, “Strongly regular graphs and spin models for the Kauffman polynomial,” Geom. Dedicata 44 (1992), 23-52.
11. F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin. 4 (1995), 103-144.
12. F. Jaeger, “Spin models for link invariants,” in Surveys in Combinatorics 1995, P. Rowlinson (Ed.), London Mathematical Society Lecture Notes Series, Cambridge University Press, 1995, Vol. 218, pp. 71-101.
13. F. Jaeger, “Towards a classification of spin models in terms of association schemes,” in Progress in Algebraic Combinatorics, E. Bannai and A. Munemasa (Eds.), Advanced Studies in Pure Math. Mathematical Society of Japan, 1996, Vol. 24, pp. 197-225.
14. F. Jaeger, “New constructions of models for link invariants,” Pacific J. Math. 176 (1996), 71-116.
15. F. Jaeger, “On four-weight spin models and their gauge transformations,” J. Alg. Combin., to appear.
16. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Alg. Combin. 8 (1998), 39-72.
17. V.F.R. Jones, “A polynomial invariant for knots via von Neumann algebras,” Bull. Am. Math. Soc. 12 (1985), 103-111.
18. V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pacific J. Math. 137 (1989), 311-336.
19. L.H. Kauffman, “State models and the Jones polynomial,” Topology 26 (1987), 395-407.
20. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications 3 (1994), 465-475.
21. K. Nomura, “Spin models constructed from Hadamard matrices,” J. Combin. Theory Ser. A 68 (1994), 251-261.
22. K. Nomura, “An algebra associated with a spin model,” J. Alg. Combin. 6 (1997), 53-58.