Knot Invariants and Iterated Integrals

We give precise formulae for the coeecients of Drin-feld's KZ associator in terms of iterated integrals over the unit interval. These formulae are used to calculate Kontsevich's universal knot invariant for (2; p)-torus knots up to the 4-th order. There are already several combinatorial descriptions of Kontsevich's universal knot invariant Ko] in terms of Drinfeld's work on quasi-triangular quasi-Hopf algebras. See B2], C], LM3] and P]. Drinfeld's work was presented in D1,2]. We only mention here that the category of representations of a quasi-triangular quasi-Hopf algebra is a tensorial category, from which one can construct framed link invariants (see e.g., AC] and RV]). The essential structure of a quasi-triangular quasi-Hopf algebra A is determined by two objects. One is an element R 2 AA, called R-matrix, which measures the non-commutativity of A and the other is an element 2 A A A, called associator, which measures the non-associativity of A. For the purpose of constructing link invariants, one can always choose R to be very simple and all the dii-culties lie in constructing. In D2], Drinfeld constructed an associator KZ using the monodromy of the formal Knizhnik-Zamolochikov connection. He also suggested a combinatorial construction which would yield an associator with rational coeecients. A detailed discussion of a combinatorial construction of such a pair (R;) appeared in B2]. Also, it was proved in LM2] that the coeecients of KZ are determined by multiple-numbers. In Section 1 of this note, we give precise formulae expressing the co-eecients of KZ as iterated integrals on the unit interval. Our calculation of the rst a few coeecients of KZ using these formulae suggests that log KZ might admit a very beautiful expression. In Section 2,