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[108] Configuration Spaces and Imbedding Invariants

  • Loring W. Tu
  • Raoul Bott
Chapter
Part of the Contemporary Mathematicians book series (CM)

Introduction

For the past two years I have been trying to understand the new physics-inspired invariants of t hree-manifolds and knots in terms of concepts more congenial to the classical topologist.

In [K] Kontsevich indicates a method for defining a set of invariants for 3-manifolds M, which presumably recreates the asymptotic invariants of the Chern-Simons theory at the trivial representation of π1(M), entirely in terms of the De Rham Theory of Configuration spaces of M. It therefore seemed plausible, ℝ3 being simply con nected, that a similar approach would work for knots in ℝ3, and in our recent paper [B-T] Cliff Taubes and I showed that this is indeed the case. Since t hen we have noted that this method yields potential invariants also for higher-dimensional knots of S2k - l in ℝ2k+l, and the first of these will be presented in section 3. How ever, in this account I will mainly concentrate on the classical case and try and fit our constructions into the picture as it emerges...

References

  1. Raoul Bott and Clifford Taubes: On the self-l inking of knots, J. Math. Phy. 35 (10) Oct. 1994Google Scholar
  2. W. Fulton and R. MacPherson: Co mpactification of configuration spaces, Ann. Math. 139, 183-225 (1994)MathSciNetCrossRefGoogle Scholar
  3. S. Axelrod and I.M. Singer: Chern-Simons perturbation theory, Proceedings of the XXth DGM Conference, edited by S. Cato and A. Rocha (World Scientific, Singapore, 1992), pp. 3-45; also Chern-Simons perturbation theory II, Preprint 1993.Google Scholar
  4. M. Kontsevich: Feynman diagrams and low-dimensional topology, Max Planck Institute Preprint, 1992.Google Scholar
  5. V.I. Arnold, The cohomology of the colored braid group, Mat. Zametki 5, 227-233 (1969).MathSciNetGoogle Scholar
  6. VD. Bar-Natan: On the Vassiliev knot invariants, Topology (1994) .Google Scholar

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Authors and Affiliations

  • Loring W. Tu
    • 1
  • Raoul Bott
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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