**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 **R**^{3}. 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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