Abstract
A new link invariant is derived using the exactly solvable chiral Potts model and a generalized Gaussian summation identity. Starting from a general formulation of link invariants using edge-interaction spin models, we establish the uniqueness of the invariant for self-dual models. We next apply the formulation to the self-dual chiral Potts model, and obtain a link invariant in the form of a lattice sum defined by a matrix associated with the link diagram. A generalized Gaussian summation identity is then used to carry out this lattice sum, enabling us to cast the invariant into a tractable form. The resulting expression for the link invariant is characterized by roots of unity and does not appear to belong to the usual quantum group family of invariants. A table of invariants for links with up to eight crossings is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. F. R. Jones,Pacific J. Math. 137:311 (1989).
M. Wadati, T. Deguchi, and Y. Akutsu,Phys. Rep. 180:247 (1989).
F. Y. Wu,Rev. Mod. Phys. 64:1099 (1992).
V. F. R. Jones,Bull. Am. Math. Soc. 12:103 (1985).
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. C. Millett, and A. Ocneanu,Bull. Am. Math. Soc. 12:239 (1985).
D. Goldschmidt and V. F. R. Jones,Geom. Dedicata 31:165 (1989); V. F. R. Jones, Commun. Math. Phys.125:459 (1989).
E. Date, M. Jimbo, K. Miki, and T. Miwa,Pacific J. Math. 154:37 (1992).
F. Jaeger,Geom. Dedicata 44:23 (1992).
P. de la Harpe and V. F. R. Jones,J. Comb. Theory B 57:207 (1993).
E. Bannai and E. Bannai,Mem. Fac. Sci. Kyushu Univ. A 47:397 (1993).
P. de la Harpe,Pacific J. Math. 162:57 (1994).
F. Y. Wu, P. Pant, and C. King,Phys. Rev. Lett. 72:3937 (1994).
H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang, and M. L. Yan,Phys. Lett. 123A:219 (1987).
R. J. Baxter, J. H. H. Perk, and H. Au-Yang,Phys. Lett A 128:138 (1988).
C. L. Siegel,Nachr. Akad. Wiss. Göttingen Math.-Phys. Klasse 1:1 (1960).
T. Kobayashi, H. Murakami and J. Murakami,Proc. Japan Acad. 64A:235 (1988).
R. J. Baxter, S. B. Kelland, and F. Y. Wu,J. Phys. A 9:397 (1976).
K. Reidemeister,Knotentheorie (Chelsea, New York, 1948).
F. Y. Wu and Y. K. Wang,J. Math. Phys. 17:439 (1976).
V. A. Fateev and A. B. Zamolodchikov,Phys. Lett. A 92:37 (1982).
F. Harary,Graph Theory (Addison-Wesley, New York, 1971).
L. Goeritz,Math. Z 36:647 (1933).
G. Burde and H. Zieschang,Knots (Walter de Gruyter, New York, 1985).
L. A. Traldi,Math. Z. 188:203 (1985).
Y. Akutsu and M. Wadati,J. Phys. Soc. Japan 56:3039 (1987).
W. B. R. Lickorish and K. C. Millett,Math. Mag. 61:3 (1988).
J. W. S. Cassels,Rational Quadratic Forms (Academic Press, London, 1972).
C. F. Gauss,Summatio quarundam serierum singularium, 1808, inWerke II (Göttingen, 1870).
A. Krazer, inFestschrift H. Weber (Leipzig u., Berlin, 1912).
M. Eichler,Quadratische Formen und Orthogonale Gruppen (Springer-Verlag, Berlin, 1952).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wu, F.Y., Pant, P. & King, C. The chiral potts model and its associated link invariant. J Stat Phys 78, 1253–1276 (1995). https://doi.org/10.1007/BF02180131
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02180131