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Knots, links, braids and exactly solvable models in statistical mechanics. (English) Zbl 0651.57005
The authors present a general method to construct isotopy invariants of classical links from exactly solvable models in statistical mechanics. They show that the Boltzmann weights of such models (which satisfy the Yang-Baxter equation) give rise to representations of the braid groups. The authors specifically consider the Boltzmann weights for the N-state vertex model proposed by K. Sogo, Y. Akutsu, T. Abe in 1983. The authors associate with the new braid group representations the so-called Markov traces and use them to derive (via the Alexander-Markov reduction of links to braids) a series of one-variable polynomial invariants of links. The polynomials corresponding to \(N=2,3,4\) are treated in some detail. The \(N=2\) polynomial is the original Jones polynomial. The other polynomials seem to be new. The authors also present a 2-variable extension of the \(N=3\) polynomial similar to the well known 2-variable extension of the Jones polynomial.
Reviewer’s remark. Essentially the same construction of the isotopy invariants of links from the Yang-Baxter matrices was developed by the reviewer [Invent. Math. 92, 527-553 (1988)].
Reviewer: V.Turaev

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B23 Exactly solvable models; Bethe ansatz
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[1] Birman, J.S.: Braids, links, and mapping class groups. Princeton, NJ: Princeton University Press 1974
[2] Rolfsen, D.: Knots and links. Berkeley, CA: Publish or Perish 1976 · Zbl 0339.55004
[3] Artin, E.: Ann. Math.48, 101 (1947) · Zbl 0030.17703 · doi:10.2307/1969218
[4] Alexander, J.W.: Proc. Natl. Acad. Sci. USA9, 93 (1923) · doi:10.1073/pnas.9.3.93
[5] Markov, A.A.: Recueil Math. Moscov 73 (1935)
[6] Jones, V.F.R.: Invent. Math.72, 1 (1983) · Zbl 0508.46040 · doi:10.1007/BF01389127
[7] Jones, V.F.R.: Bull. Am. Math. Soc.12, 103 (1985) · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2
[8] Temperley, H.N.V., Lieb, E.H.: Relations between the ?percolation? and ?colouring? problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the ?percolation? problem. Proc. Roy. Soc. Lond. A322, 251 (1971) · Zbl 0211.56703
[9] Lieb, E.H., Wu, F.Y.: In: Phase transitions and critical phenomena, Vol. 1, p. 331. Domb, C., Green, M.S. (eds.). London: Academic Press 1972
[10] Baxter, R.J., Kelland, S.B., Wu, F.Y.: Equivalence of the Potts’ model or Whitney polynomial with an ice-type model. J. Phys. A9, 397 (1976) · Zbl 0321.05140
[11] Bourbaki, N.: Groupes et algebres de Lie. Paris: Hermann 1968, Chap. 4 · Zbl 0186.33001
[12] Alexander, J.W.: Trans. Am. Math. Soc.30, 275 (1928) · doi:10.1090/S0002-9947-1928-1501429-1
[13] Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: Bull. Am. Math. Soc.12, 239 (1985) · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3
[14] Birman, J.S.: Invent. Math.81, 287 (1985) · Zbl 0588.57005 · doi:10.1007/BF01389053
[15] Kanenobu, T.: Math. Ann.275, 555 (1986) · Zbl 0594.57005 · doi:10.1007/BF01459137
[16] Faddeev, L.D.: Sov. Sci. Rev. Math. Phys. C1, 107 (1981)
[17] Thacker, H.B.: Exact integrability in quantum field theory and statistical mechanics. Rev. Mod. Phys.53, 253 (1981) · doi:10.1103/RevModPhys.53.253
[18] Kulish, P.P., Sklyanin, E.K.: Lecture Notes in Physics, Vol. 151, p. 61. Berlin, Heidelberg, New York: Springer 1982
[19] Wadati, M.: In: Dynamical problems in soliton systems, p. 68. Takeno, S. (ed.). Berlin, Heidelberg, New York: Springer 1985
[20] Wadati, M., Akutsu, Y.: Exactly solvable models in statistical mechanics. In: Springer Series in Nonlinear Dynamics. Lakshmanan, M. (ed.). Berlin, Heidelberg, New York: Springer 1988 · Zbl 0669.35105
[21] Karowski, M., Thun, H.J., Truong, T.T., Weisz, P.H.: On the uniqueness of a purely elasticS-matrix in (1+1) dimensions. Phys. Lett.67, 321 (1977)
[22] Zamolodchikov, A.B., Zamolodchikov, A.B.: FactorizedS-Matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. (NY)120, 253 (1979) · Zbl 0946.81070 · doi:10.1016/0003-4916(79)90391-9
[23] Sogo, K., Uchinami, M., Nakamura, A., Wadati, M.: Nonrelativistic theory of factorizedS-matrix. Prog. Theor. Phys.66, 1284 (1981) · Zbl 1074.81583 · doi:10.1143/PTP.66.1284
[24] Sogo, K., Uchinami, M., Akutsu, Y., Wadati, M.: Classification of exactly solvable two-component models. Prog. Theor. Phys.68, 508 (1981) · Zbl 1073.82546 · doi:10.1143/PTP.68.508
[25] Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press 1982 · Zbl 0538.60093
[26] Wu, F.Y.: Ising model with four-spin interactions. Phys. Rev. B4, 2312 (1971)
[27] Kadanoff, L.P., Wegner, F.J.: Some critical properties of the eight vertex model. Phys. Rev. B4, 3989 (1981)
[28] Zamolodchikov, A.B.:Z 4-symmetric factorizedS-matrix in two space-time dimensions. Commun. Math. Phys.69, 165 (1979) · doi:10.1007/BF01221446
[29] Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys.35, 193 (1984) · Zbl 0589.60093 · doi:10.1007/BF01014383
[30] Kuniba, A., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn55, 1092, 2170, and 3338 (1986) · doi:10.1143/JPSJ.55.1092
[31] Kuniba, A., Akutsu, Y., Wadati, M.: An exactly solvable 4-state IRF model. Phys. Lett.116, 382 (1986) and An exactly solvable 5-state IRF model.117 A, 358 (1986) · doi:10.1016/0375-9601(86)90060-5
[32] Baxter, R.J., Andrews, G.E.: Lattice gas generalization of the hard hexagon model. I. Startriangle relation and local densities. J. Stat. Phys.44, 249 (1986) · Zbl 0638.10007 · doi:10.1007/BF01010916
[33] Andrews, G.E., Baxter, R.J.: Lattice gas generalization of the hard hexagon model. II. The local densities as elliptic functions. J. Stat. Phys.44, 713 (1986) · Zbl 0638.10008 · doi:10.1007/BF01011904
[34] Akutsu, Y., Kuniba, A., Wadati, M.: J. Phys. Soc. Jpn.55, 1466 and 1880 (1986) · doi:10.1143/JPSJ.55.1466
[35] Kuniba, A., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.55, 2605 (1986) · doi:10.1143/JPSJ.55.2605
[36] Akutsu, Y., Kuniba, A., Wadati, M.: J. Phys. Soc. Jpn.55, 290 (1986)
[37] Date, E., Jimbo, M., Miwa, T., Okado, M.: Fusion of the eight vertex SOS model. Lett. Math. Phys.12, 209 (1986) · doi:10.1007/BF00416511
[38] Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.56, 839 (1987) · Zbl 0719.57002 · doi:10.1143/JPSJ.56.839
[39] Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.56, 3039 (1987) · Zbl 0719.57003 · doi:10.1143/JPSJ.56.3039
[40] Sogo, K., Akutsu, Y., Abe, T.: New factorizedS-matrix and its application to exactly solvableq-state model. I and II. Theor. Phys.70, 730 and 739 (1983) · Zbl 1098.82548
[41] Zamolodchikov, A.B., Fateev, V.A.: A model of facterizedS-matrix and an integrable spin-1 Heisenberg chain. Sov. J. Nucl. Phys.32, 298 (1980)
[42] Powers, R.T.: Ann. Math.86, 138 (1967) · Zbl 0157.20605 · doi:10.2307/1970364
[43] Pimsner, M., Popa, S.: Preprint
[44] Conway, J.H.: In: Computational problems in abstract algebra, p. 329. Leach, J. (ed.). London: Pergamon Press 1969 · Zbl 0186.19802
[45] Akutsu, Y., Deguchi, T., Wadati, M.: J. Phys. Soc. Jpn.56, 3464 (1987) · Zbl 0719.57004 · doi:10.1143/JPSJ.56.3464
[46] Deguchi, T., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.57, No. 3 (1988)
[47] Kauffman, L.H.: Preprint
[48] Birman, J.S., Wenzl, H.: Preprint
[49] Murakami, J.: Preprint
[50] Murakami, J.: Preprint
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