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 V x containing elements g i satisfying the usual braid group relations and elements a i satisfying g i−g sup-1infi =εa i, where ε is a formal variable that may be regarded as measuring the failure of g sup2infi to equal 1. Topologically, the elements a i 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 V x. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.
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References
AshtekarA., New Hamiltonian formulation of general relativity, Phys. Rev. D36, 1587–1602 (1987).
Baez, J., Quantum gravity and the algebra of tangles, UC Riverside preprint, 1992.
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.
BirmanJ., Braids, Links, and Mapping Class Groups, Princeton University Press, Princeton, 1974.
Birman, J. and Lin, X.-S., Knot polynomials and Vassiliev's invariants, Columbia University preprint, 1991.
BrügmannB., GambiniR., and PullinJ., Knot invariants as nondegenerate quantum geometries, Phys. Rev. Lett. 68, 431–434 (1992).
Brügmann, B., Gambini, R., and Pullin, J., Jones polynomials for intersecting knots as physical states for quantum gravity, University of Utah preprint, 1992.
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).
Gambini, R., Link invariant polynomials for intersecting loops, IFFI preprint, 1992.
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).
KauffmanL., State models for link polynomials, Enseign. Math. 36, 1–37 (1990).
KodamaH., Holomorphic wave function of the universe, Phys. Rev. D42, 2548–2565 (1990).
Lin, X.-S., Vertex models, quantum groups and Vassiliev's knot invariants, Columbia University preprint, 1991.
RovelliC. and SmolinL., Loop representation for quantum general relativity, Nuclear Phys. B331, 80–152 (1990).
SmolinL., Invariants of links and critical points of the Chern-Simons path integral, Modern Phys. Lett. A4, 1091–1112 (1989).
SmolinL., The G Newton → 0 limit of Euclidean quantum gravity, J. Classical Quantum Gravity 9, 883–893 (1992).
Stanford, T., Finite type invariants of knots, links and graphs, Columbia University preprint, 1992.
TuraevV., The Yang-Baxter equation and invariants of links, Invent. Math. 92, 527–553 (1988).
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.
WittenE., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, 351–399 (1989).