Letters in Mathematical Physics

, Volume 26, Issue 1, pp 43–51 | Cite as

Link invariants of finite type and perturbation theory

  • John C. Baez
Article

Abstract

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra Vx containing elements gi satisfying the usual braid group relations and elements ai satisfying gig infi sup-1 ai, where ε is a formal variable that may be regarded as measuring the failure of g infi sup2 to equal 1. Topologically, the elements ai signify intersections. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on Vx. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.

Mathematics Subject Classifications (1991)

81T13 57M25 20F36 

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References

  1. 1.
    AshtekarA., New Hamiltonian formulation of general relativity, Phys. Rev. D36, 1587–1602 (1987).Google Scholar
  2. 2.
    Baez, J., Quantum gravity and the algebra of tangles, UC Riverside preprint, 1992.Google Scholar
  3. 3.
    Bar-Natan, D., Weights of Feynman diagrams and the Vassiliev knot invariants, Princeton University preprint, 1991; Bar-Natan, D., Isotopy invariance for knots in perturbative Chern-Simons theory, Harvard University preprint, 1992.Google Scholar
  4. 4.
    BirmanJ., Braids, Links, and Mapping Class Groups, Princeton University Press, Princeton, 1974.Google Scholar
  5. 5.
    Birman, J. and Lin, X.-S., Knot polynomials and Vassiliev's invariants, Columbia University preprint, 1991.Google Scholar
  6. 6.
    BrügmannB., GambiniR., and PullinJ., Knot invariants as nondegenerate quantum geometries, Phys. Rev. Lett. 68, 431–434 (1992).Google Scholar
  7. 7.
    Brügmann, B., Gambini, R., and Pullin, J., Jones polynomials for intersecting knots as physical states for quantum gravity, University of Utah preprint, 1992.Google Scholar
  8. 8.
    Cotta-RamusinoP., GuadagniniE., MartelliniM., and MintchevM., Quantum field theory and link invariants, Nuclear Phys. B330, 557–574 (1990); Guadagnini, E., Martellini, M., and Mintchev, M., Wilson lines in Chern-Simons theory and link invariants, Nuclear Phys. B330, 575–607 (1990).Google Scholar
  9. 9.
    Gambini, R., Link invariant polynomials for intersecting loops, IFFI preprint, 1992.Google Scholar
  10. 10.
    GusarovM., A new form of the Conway-Jones polynomial of oriented links, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 193, 4–9 (1991).Google Scholar
  11. 11.
    KauffmanL., State models for link polynomials, Enseign. Math. 36, 1–37 (1990).Google Scholar
  12. 12.
    KodamaH., Holomorphic wave function of the universe, Phys. Rev. D42, 2548–2565 (1990).Google Scholar
  13. 13.
    Lin, X.-S., Vertex models, quantum groups and Vassiliev's knot invariants, Columbia University preprint, 1991.Google Scholar
  14. 14.
    RovelliC. and SmolinL., Loop representation for quantum general relativity, Nuclear Phys. B331, 80–152 (1990).Google Scholar
  15. 15.
    SmolinL., Invariants of links and critical points of the Chern-Simons path integral, Modern Phys. Lett. A4, 1091–1112 (1989).Google Scholar
  16. 16.
    SmolinL., The G Newton → 0 limit of Euclidean quantum gravity, J. Classical Quantum Gravity 9, 883–893 (1992).Google Scholar
  17. 17.
    Stanford, T., Finite type invariants of knots, links and graphs, Columbia University preprint, 1992.Google Scholar
  18. 18.
    TuraevV., The Yang-Baxter equation and invariants of links, Invent. Math. 92, 527–553 (1988).Google Scholar
  19. 19.
    VassilievV., Cohomology of knot spaces, in V. I.Arnold (ed), Theory of Singularities and Its Applications, Advances in Soviet Math. Vol. 1, Amer. Math. Soc., Providence, 1990.Google Scholar
  20. 20.
    WittenE., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, 351–399 (1989).Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • John C. Baez
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA

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