Abstract
We show how the Conway Alexander polynomial arises from theq deformation of (Z 2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU q> sl(1, 1) algebra describes free fermions “propagating” on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.
Similar content being viewed by others
References
[A23] Alexander, J.W.: Trans. Am. Math. Soc.30, 275–306 (1923)
[AW87] Akutsu, Y., Wadati, W.: J. Phys. Soc. Jpn.56, 839–842 (1987); J. Phys. Soc. Jpn.56, 3039–3051 (1987)
[BB80] Balantehin, B., Bars, I.: J. Math. Phys.22 (1981)
[B74] Birman, J.S.: Braids, links and mapping class groups. Ann. Math. Study82 (1974), Princeton Univ. Press
[C70] Conway, J.H.: In: Computational problems in abstract algebra. pp. 329–358. New York: Pergamon Press 1970
[CK90] Chaichian, M., Kulish, P.P.: Phys. Lett. B234, 72 (1990)
[CO89] Corrigan, E., Fairlie, D.B., Fletcher, P., Sasaki, R.: J. Math. Phys.
[D89] Deguchi, T.: J. Phys. Soc. Jpn.58, 3441 (1989)
[D86] Drinfeld, V.G.: Quantum groups. Proc. Intl. Congress Math., Berkeley 789 (1986)
[DA90] Deguchi, T., Akutsu, Y.: J. Phys. A23, 1861 (1990)
[DJ81] Dandi, P.H., Jarvis, P.B.: J. Phys. A14, 547 (1981)
[F63] Fox, R.H., Crowell, R.H.: Introduction to Knot Theory. Blaisdell 1963
[Ja87] Jaeger, F.: Composition products and models for the homfly polynomial. Preprint 87
[J87] Jimbo, M.: Lett. Math. Phys.10, 63 (1985)
[J86] Jimbo, M.: Commun. Math. Phys.102, 537 (1986)
[Jo85] Jones, V.F.R.: Bull. Am. Math. Soc.12, 102–112 (1985)
[Jo87] Jones, V.F.R.: Ann. Math.126, 335–388 (1987)
[Jo89] Jones, V.F.R.: Pacific J. Math.137, 189 311–334 (1989)
[JY82] Joyce, D.: J. Pure Appl. Algebra23, 37–65 (1982)
[Ka77] Kac, V.G.: Commun. Math. Phys.53, 31 (1977)
[K83] Kauffman, L.H.: Formal Knot Theory. Princeton University Press Math. Notes30 (1983)
[K1, 87] Kauffman, L.H.: On Knots. Ann. Math. Study15, Princeton University Press 1987
[K2, 87] Kauffman, L.H.: Topology26, 395–407 (1987)
[K88] Kauffman, L.H.: Contemp. Math., Vol. 78. Am. Math. Soc. pp. 263–297 (1988), and New problems, methods and techniques in quantum field theory and statistical mechanics, pp. 175–222. Singapore: World Scientific 1990
[K1, 89] Kauffman, L.H.: Contemp. Math.96. Am. Math. Soc. 221–231 (1989)
[K2, 89] Kauffman, L.: Knots, abstract tensors, and the Yang Baxter equation. In: Knots topology and quantum field theories, pp. 179–334. Singapore: World Scientific 1989
[K90] Kauffman, L.H.: Trans. Am. Math. Soc.318 (2), 417–471 (1990)
[K91] Kauffman, L.H.: Knots and Physics. Singapore: World Scientific 1991
[KR89] Kulish, P.P., Reshetikhin, N.Yu.: Lett. Math. Phys.18, 143 (1989)
[KS82] Kulish, P.P., Sklyanin, E.K.: J. Sov. Math.19, 1596 (1982)
[KS91] Kauffman, L., Saleur, H.: Fermions and link invariants. Preprint YCTP-P21-91
[LCS88] Lee, H.C., Couture, M., Schmeling, N.C.: Connected link polynomials. Preprint 88
[LiM87] Lickorish, W.B.R., Millett, K.: Topology26, 107 (1987)
[LW71] Lieb, E., Wu, F.Y.: In: Phase transitions and critical phenomena, Vol. 1. Domb, C., Green, M.S. (eds.). New York: Academic Press 1971
[M62] Milnor, J.W.: Ann. Math.76, 137–147 (1962)
[PS90] Pasquier, V., Saleur, H.: Nucl. Phys. B330, 523 (1990)
[R87] Reshetikhin, N.Y.: Quantized universal enveloping algebras. Preprint, LOMI E-4-87, E-17-87, Steklov Institute, Leningrad
[R89] Reshetikhin, N.Yu.: Unpublished
[S89] Saleur, H.: Symmetries of the XX chain and applications. In: Proceedings of recent developments in conformal field theories. Trieste, Oct. 2-Oct. 4, 1989. Singapore: World Scientific 1990
[S90] Saleur, H.: Nucl. Phys. B336, 363 (1990)
[Sa80] Samuel, S.: J. Math. Phys.21, 2806 (1980)
[Sc79] Scheunert, M.: The theory of Lie superalgebras. Lect. Notes in Mathematics, Vol.716. Berlin, Heidelberg, New York: Springer 1979
[SC81] Schultz, C.L.: Phys. Rev. Lett.46, 629 (1981)
[SVZ89] Schmidke, W., Vokos, S.P., Zumino, B.: Preprint UCB-PTH-89/32
[T88] Turaev, V.G.: Invent. Math.92, 527–553 (1988)
[WDA89] Wadati, M., Deguchi, T., Akutsu, Y.: Phys. Rep.180, 247 (1989)
[W89] Witten, E.: Commun. Math. Phys.121, 351–399 (1989)
Author information
Authors and Affiliations
Additional information
Communicated by K. Gawedzki
Work supported in part by NSF grant no. DMS-8822602
Work supported in part by the NSF: grant nos. PYI PHY 86-57788 and PHY 90-00386 and by CNRS, France
Rights and permissions
About this article
Cite this article
Kauffman, L.H., Saleur, H. Free fermions and the Alexander-Conway polynomial. Commun.Math. Phys. 141, 293–327 (1991). https://doi.org/10.1007/BF02101508
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02101508