Formal descriptions of Turaev's loop operations

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra Q[pi] of the fundamental group of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra Q[pi] is completed with respect to powers of its augmentation ideal and is appropriately identified to the degree-completion of the tensor algebra T(H) of H := H-1 (pi ; Q) In this paper, we obtain a similar algebraic description for Turaev's self-intersection map in the case of a disk with p punctures. Here we consider the identification between the completions of Q[pi] and T(H) that arises from a Drinfeld associator by embedding into the pure braid group on (p + 1) strands; our algebraic description involves a formal power series which is explicitly determined by the associator. The proof is based on some three-dimensional formulas for Turaev's loop operations, which involve 2-strand pure braids and are shown for any surface with boundary.