Skip to main content
SpringerLink
Log in

Ribbon graphs and their invaraints derived from quantum groups

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The generalization of Jones polynomial of links to the case of graphs inR 3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. [Br] Brejxxxxxxxxxxxer, M.R., Moody, R.V., Patera, J.: Tables of dominant weight multiplicites for representation of simple Lie groups. New York: Dekker 1984.

    Google Scholar 

  2. [D] Deling, P., Milne, J.S.: Tannakian Categories in Hodge cicles motives and simura varieties. Lecture Notes ion Mathematics, vol 900. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  3. [Dr1] Drinfeld, V.G.: Proceedings of the International Congres of Mathematics, vol. 1, p. 798–820. Berkeley, California: New York: Academic Press 1986)

    Google Scholar 

  4. [Dr2] Drinfeld, V.G.: Quasicommutative Hopf algebras. Algebra and Anal.1 (1989)

  5. [FRT] Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantized Lie Groups and Lie algebras. LOMI-preprint, E-14-87, 1987, Leningrad

  6. [FY] Freyd, P.J., Yetter, D.N.: Braided compact closed categories with applications to low dimensional topology. Preprint (1987)

  7. [JS] Joyal, A., Street, R.: Braied Manoidal categories. Macquarie Math. Reports. Report No. 860081 (1986)

  8. [JO] Jones, V.F.R.: A polynomial invariant of Knots via von Neumann algebras. Bull. Am. Math. Soc.12, 103–111 (1985)

    Google Scholar 

  9. [KT] Tsuchiya, A., Kanlie, Y.: Vertex operators in the conformal field theory of onP1 and monodromy representations of the braid group. In: Conformal field theory and solvable lattice models. Lett. Math. Phys.13, 303 (1987)

  10. [KV] Kauffman, L.H., Vogel, P.: Link polynomials and a graphical calculus. Preprint (1987)

  11. [K] Kirillov, A.N.:q-analog of Clebsch-Gordon coeffcients for slz. Zap. Nauch. Semin. LOMI,170, 1988

  12. [KR] Kirillov, A.N., Reshetikhin, N.Yu.: Representation of the algebraU q(sl 2),q-orthogonal polynomials and invariants of links. LOMI-preprint, E-9-88, Leningrad 1988

  13. [L] Luztig, G.: Quantum derformations of certain simple modules over enveloping algebras. M.I.T. Preprint, December 1987

  14. [Ma] MacLane, S.: Natural associativity and commutavity. Rice Univ. Studies49, 28–46 (1963)

    Google Scholar 

  15. [MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989)

    Google Scholar 

  16. [Re1] Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links. I. LOMI-preprint E-4-87 (1988), II. LOMI-preprint E-17-87 (1988)

  17. [Re2] Reshetikhin, N.Yu.: Quasitriangluar Hopfalgebras and invariants of Tangles. Algebra and Anal.1, (1988) (in Russian)

  18. [RS] Reshetikhin, N.Yu.: Semenov: Tian-Shansky, M.A.: QuantumR-matrices and factorization problem in quatum groups. J. Geom. Phys. v. V (1988)

  19. [RO] Rosso, M.: Représentations irréductibles de dimension finie duq-analogue de l'algébre enveloppante d'une algèbre de Lie simple. C.R. Acad. Sci. paris.305, Se'rie 1, 587–590 (1987)

    Google Scholar 

  20. [Tu1] Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math.92, 527–553 (1988)

    Google Scholar 

  21. [Tu2] Turaev, V.G.: The Conways and Kauffman modules of the solid torus with an appendix on the operator invariants of tangles. LOMI Preprint E-6-88. Leningrad (1988)

  22. [Ye] Yetter, D.N.: Category theoretic representations of Knotted graphs in S3, Preprint

  23. [Wi] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. (in press)

Download references

Author information

Authors and Affiliations

  1. L.O.M.I., Fontanka 27, SU-191001, Leningrad, USSR

    N. Y. Reshetikhin & V. G. Turaev

Authors
  1. N. Y. Reshetikhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Turaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reshetikhin, N.Y., Turaev, V.G. Ribbon graphs and their invaraints derived from quantum groups. Commun.Math. Phys. 127, 1–26 (1990). https://doi.org/10.1007/BF02096491

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096491

Keywords

Search

Navigation