On Four-Weight Spin Models and their Gauge Transformations
François Jaeger
DOI: 10.1023/A:1008778103812
Abstract
We study the four-weight spin models (W 1, W 2, W 3, W 4) 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 (W 1, W 3), or the same pair (W 2, W 4). 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 W 1, W 3 or preserve W 2, W 4. Such 
gauge transformations 
correspond to multiplication of W 2, W 4 by permutation matrices representing certain symmetries of the spin model, and to conjugation of W 1, W 3 by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W 1, W 3 are symmetric.
gauge transformations correspond to multiplication of W 2, W 4 by permutation matrices representing certain symmetries of the spin model, and to conjugation of W 1, W 3 by diagonal matrices. We show for instance that up to gauge transformations, we can assume that W 1, W 3 are symmetric.Pages: 241–268
Keywords: spin model; link invariant; association scheme; Bose-mesner algebra
Full Text: PDF
References
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2. E. Bannai and E. Bannai, “Generalized spin models (four-weight spin models),” Pacific J. of Math. 170 (1995), 1-16.
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4. Ei. Bannai and T. Ito, Algebraic Combinatorics, Vol. I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.
5. Et. Bannai and A. Munemasa, “Duality maps of finite abelian groups and their applications to spin models,” preprint, 1995.
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11. D.M. Goldschmidt and V.F.R. Jones, “Metaplectic link invariants,” Geom. Dedicata 31 (1989), 165-191.
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13. F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin. 4 (1995), 103-144.
14. E. Jaeger, “Towards a classification of spin models in terms of association schemes,” Progress in Algebraic Combinatorics, Mathematical Society of Japan, 1996, pp. 197-225. Advanced Studies in Pure Mathematics, Vol. 24.
15. F. Jaeger, “New constructions of models for link invariants,” Pacific J. Math. 176 (1996), 71-116.
16. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Algebraic Combin. 8 (1998), 39-72. JAEGER
17. V.F.R. Jones, “A polynomial invariant for knots via Von Neumann algebras,” Bull. Am. Math. Soc. 12 (1985), 103-111.
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22. W.B.R. Lickorish and K. Millett, “A polynomial invariant of oriented links,” Topology 26 (1987), 107-141.
23. H.R. Morton and P.R. CROMWELL, “Distinguishing mutants by knot polynomials,” J. of Knot Theory and its Ramifications 5 (1996), 225-238.
24. K. Nomura, “An algebra associated with a spin model,” preprint, 1994.
25. N. Yu. Reshetikhin and V.G. Turaev, “Ribbon graphs and their invariants derived from quantum groups,” Commun. Math. Phys. 127 (1990), 1-26.
26. V.G. Turaev, “The Yang-Baxter equation and invariants of links,” Invent. Math. 92 (1988), 527-553.
27. F.Y. Wu, P. Pant, and C. King, “New link invariant from the chiral Potts model,” Phys. Rev. Lett. 72 (1994), 3937-3940.
28. M. Yamada, “The construction of four-weight spin models by using Hadamard matrices and M-structure,” Australasian J. of Comb. 10 (1994), 237-244.
2. E. Bannai and E. Bannai, “Generalized spin models (four-weight spin models),” Pacific J. of Math. 170 (1995), 1-16.
3. E. Bannai, E. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,” J. Algebraic Combin. 6 (1997), 203-228.
4. Ei. Bannai and T. Ito, Algebraic Combinatorics, Vol. I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.
5. Et. Bannai and A. Munemasa, “Duality maps of finite abelian groups and their applications to spin models,” preprint, 1995.
6. A.E. Brouwer, A.M. Cohen, and A. Neumaier, “Distance-Regular Graphs, Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete,
3. Folge, Band 18, 1989.
7. G. Burde and H. Zieschang, Knots, de Gruyter, Berlin, New York, 1985.
8. R.H. Crowell and R.H. Fox, Introduction to Knot Theory, Springer, Berlin, 1963.
9. E. Date, M. Jimbo, K. Miki, and T. Miwa, “Braid group representations arising from the generalized chiral Potts models,” Pacific J. of Math. 154(1) (1992), 37-66.
10. T. Deguchi, “Generalized spin models associated with exactly solvable models,” Progress in Algebraic Combinatorics, Mathematical Society of Japan, 1996, pp. 81-100. Advanced Studies in Pure Mathematics, Vol. 24.
11. D.M. Goldschmidt and V.F.R. Jones, “Metaplectic link invariants,” Geom. Dedicata 31 (1989), 165-191.
12. F. Jaeger, “Spin models for link invariants,” in Surveys in Combinatorics 1995, Peter Rowlinson (Ed.), Cambridge University Press 1995, pp. 71-101. London Mathematical Society Lecture Notes Series, Vol. 218.
13. F. Jaeger, “On spin models, triply regular association schemes, and duality,” J. Alg. Combin. 4 (1995), 103-144.
14. E. Jaeger, “Towards a classification of spin models in terms of association schemes,” Progress in Algebraic Combinatorics, Mathematical Society of Japan, 1996, pp. 197-225. Advanced Studies in Pure Mathematics, Vol. 24.
15. F. Jaeger, “New constructions of models for link invariants,” Pacific J. Math. 176 (1996), 71-116.
16. F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matrices and spin models,” J. Algebraic Combin. 8 (1998), 39-72. JAEGER
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. of Math. 137 (1989) 2, 311-334.
19. V.G. Kac and M. Wakimoto, “A construction of generalized spin models,” Perspectives in Mathematical Physics, International Press, 1994, pp. 125-150. Conf. Proc. Lecture Notes Math. Phys. III.
20. L.H. Kauffman, On Knots, Princeton University Press, Princeton, New Jersey,
1987. Annals of Mathematical Studies, Vol. 115.
21. K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theory and its Ramifications 3 (1994), 465-476.
22. W.B.R. Lickorish and K. Millett, “A polynomial invariant of oriented links,” Topology 26 (1987), 107-141.
23. H.R. Morton and P.R. CROMWELL, “Distinguishing mutants by knot polynomials,” J. of Knot Theory and its Ramifications 5 (1996), 225-238.
24. K. Nomura, “An algebra associated with a spin model,” preprint, 1994.
25. N. Yu. Reshetikhin and V.G. Turaev, “Ribbon graphs and their invariants derived from quantum groups,” Commun. Math. Phys. 127 (1990), 1-26.
26. V.G. Turaev, “The Yang-Baxter equation and invariants of links,” Invent. Math. 92 (1988), 527-553.
27. F.Y. Wu, P. Pant, and C. King, “New link invariant from the chiral Potts model,” Phys. Rev. Lett. 72 (1994), 3937-3940.
28. M. Yamada, “The construction of four-weight spin models by using Hadamard matrices and M-structure,” Australasian J. of Comb. 10 (1994), 237-244.
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