Configuration Spaces and Imbedding Invariants
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...
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