Chord diagram invariants of tangles and graphs

The notion of a chord diagram emerged from Vassiliev's work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This knot invariant establishes an isomorphism of a projective limit of algebras generated by the Vassiliev equivalence classes of knots onto the algebra of chord diagrams. Kontsevich originally deened his invariant of knots via a multiple integral given by an explicit but complicated analytic expression. This expression, however beautiful, does not reveal the combinatorial nature of the invariant. (A similar situation would occur if the linking number of knots were introduced via the Gauss integral formula without a combinatorial calculation). A combinatorial reformulation of the Kontsevich integral appeared in the works of Bar-Natan BN94], Cartier Car93], Le and Murakami LM93], Piunikhin Piu95] (see also Kas95, Chapter XX]). On the algebraic side, this reformulation uses the notions of braided and innnitesimal symmetric categories as well as the notion of an associator introduced by Drinfeld Dri89] in his study of quasitriangular quasi-Hopf algebras. On the geometric side, one uses categories of tangles, as introduced by Yetter and Turaev (see Tur94]). Note also that a counterpart of the Kontsevich knot invariant in the theory of braids was discovered earlier by Kohno Koh85] who considered an algebraic version of chord diagrams. In this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we x a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coeecients in R, cf. Section 2. The set of morphisms in T (R) has a canonical ltration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I m+1 are exactly the Vassiliev invariants of degree m for framed oriented tangles. Completing T (R) at the ideal I, we obtain the pro-unipotent completion b T (R) = lim ?m T (R)=I m+1. Our main result (Corollary 2.5) states that, if R contains the eld Q of rational numbers, then b T (R) is isomorphic to a suitable completion b A(R) of the category …