Duke Math. J. 92 (1998), 497-552

Christian Kassel and Vladimir Turaev

Chord diagram invariants of tangles and graphs

Mathematics Subject Classification (1991): 17B37, 18D10, 19D23, 57M25, 81R50

Abstract. We extend Kontsevich's universal knot invariant to embedded graphs in R3. Our construction relies on several categorical concepts and the following facts:

(i) The free infinitesimal symmetric category on one object is a category whose sets of morphisms are spanned by the chord diagrams that appear in the theory of Vassiliev invariants.

(ii) We prove that the pro-unipotent completion of the tangle category is isomorphic to the above-mentioned category of chord diagrams. This is a categorical generalization of Kontsevich's isomorphism and it allows us to compute the quantum invariants of a knot from its chord diagram invariant.

Using Drinfeld's work on the Grothendieck-Teichmueller group, we also construct an action of the Galois group Gal(Q-/Q) on the Vassiliev invariants of knots and links.

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