Abstract
First we briefly recall the definition of the three-dimensional Baxter-Bazhanov lattice model. The spins of this model are elements ofZ N and theR-matrix is associated to the algebraU q sl(n) ifq is a primitiveNth root of unity. Then we construct a particularN→∞ limit of the model, in which it is meaningful to interpret the spins as elements ofR and which gives the free Gaussian boson model. Finally, we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of Alexander polynomials, evaluated at certain roots of unity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. V. Bazhanov and R. J. Baxter, New solvable lattice models in three dimensions,J. Stat. Phys. 69:453 (1992).
A. B. Zamolodchikov, Tetrahedra equations and integrable systems in three-dimensional space,Zh. Eksp. Teor. Fiz. 79:641 (1980) [Sov. Phys. JETP 52:325 (1980)].
A. B. Zamolodchikov, Tetrahedron equations and the relativistic S-matrix of straightstrings in 2+1-dimensions,Commun. Math. Phys. 79:489 (1981).
V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov,Z n−1 N generalization of the chiral Potts model,Commun. Math. Phys. 138:393 (1991).
E. Date, M. Jimbo, K. Miki, and T. Miwa, Generalized chiral Potts models and minimal cyclic representations ofU q (ĝl(n, C)),Commun. Math. Phys. 137:133 (1991).
R. M. Kashaev, V. V. Mangazeev, and T. Nakanishi, Yang-Baxter equation for thesl(n) chiral Potts model,Nucl. Phys. B 362:563 (1991).
B. L. Cerchiai, M. Martellini, and F. Valz-Gris, Three-dimensional integrable models and associated tangle invariants,J. Stat. Phys. 78:1083 (1994).
E. Date, M. Jimbo, K. Miki, and T. Miwa, Braid group representations arising from the generalized chiral Potts models,Pac. J. Math. 154:37 (1992).
T. Kobaiashi, H. Murakami, and J. Murakami, Cyclotomic invariants for links,Proc. Jpn. Acad. 64A:235 (1988).
V. R. Jones, Baxterization,Int. J. Mod. Phys. A 6:2035 (1991).
D. Goldschmidt and V. R. Jones, Metaplectic link invariants,Geom. Dedicata 31:165 (1989).
V. V. Bazhnov and R. J. Baxter, Star-triangle relation for a three dimensional model,J. Stat. Phys. 71:839 (1993).
V. Turaev,Russ. Math. Surv. 41:199 (1986).
R. J. Baxter, On Zamolodchikov's solution of the tetrahedron equations,Commun. Math. Phys. 88:185 (1983).
R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov, Spatial symmetry, local integrability and tetrahedron equations in the Baxter-Bazhanov model,Int. J. Mod. Phys. A 8:587 (1993).
R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov, Star-square and tetrahedron equations in the Baxter-Bazhanov model,Int. J. Mod. Phys. A 8:1399 (1993).
T. Deguchi, M. Wadati, and Y. Akutsu, Knot theory based on solvable models at criticality,Adv. Stud. Pure Math. 19:193 (1989).
M. Wadati, T. Deguchi and Y. Akutsu, Exactly solvable models and knot theory,Phys. Rep. 180:247 (1989).
V. A. Fateev and A. B. Zamolodchikov, Self-dual solutions of the star-triangle relations inZ N -models,Phys. Lett. 92A:37 (1982).
V. A. Fateev and A. B. Zamolodchikov, The exactly solved case of a 2D lattice of plane rotators,Phys. Lett. 92A:35 (1982).
D. Gepner,Nucl. Phys. B 290[FS20]:10 (1987).
L. Rozansky, A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds, II, preprint UMTG-176-94, hepth/9403021 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cerchiai, B.L., Cotta-Ramusino, P. & Martellini, M. Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model. J Stat Phys 81, 629–645 (1995). https://doi.org/10.1007/BF02179250
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179250