Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra 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 is completed with respect to powers of its augmentation ideal and is appropriately identified to the degree-completion of the tensor algebra of .
In this paper, we obtain a similar algebraic description for Turaev's self-intersection map in the case of a disk with punctures. Here we consider the identification between the completions of and that arises from a Drinfeld associator by embedding into the pure braid group on 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 -strand pure braids and are shown for any surface with boundary.
Cite this article
Gwénaël Massuyeau, Formal descriptions of Turaev's loop operations. Quantum Topol. 9 (2018), no. 1, pp. 39–117DOI 10.4171/QT/103