Abstract:
Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[\pi]$ of the fundamental group $\pi$ of a surface with boundary [1].
The first operation is binary and measures the intersection of two oriented curves on the surface,
while the second operation is unary and computes the self-intersection of an oriented curve.
It is already known that Turaev's intersection pairing has a simple algebraic description
when the $I$-adic completion of the group algebra $\mathbb{Q}[\pi]$ is appropriately identified to the degree-completion of the tensor algebra $T(H)$ of $H:=H_1(\pi;\mathbb{Q})$.
We will show that Turaev's self-intersection map has a similar description in the case of a disk with $p$ punctures.
In this special case, we will consider those identifications between the completions of $\mathbb{Q}[\pi]$ and $T(H)$
that arise from the Kontsevich integral by embedding $\pi$ into the pure braid group on $(p+1)$ strands [2, 3].
As a matter of fact, our algebraic description involves a formal power series which is explicitly determined by the Drinfeld associator $\Phi$
entering into the definition of the Kontsevich integral; this series is essentially Enriquez' $\Gamma$-function of $\Phi$ [4].
If time allows, we will also discuss the case of higher-genus surfaces. (This talk is based on the preprint [5].)
References:
V. Turaev, Intersections of loops in two-dimensional manifolds. (Russian) Mat. Sb. 106(148) (1978),
no. 4, 566–588. English translation: Math. USSR–Sb. 35 (1979), 229–250.
N. Habegger, G. Masbaum, The Kontsevich integral and Milnor's invariants. Topology 39 (2000), no. 6, 1253–1289.
A. Alekseev, B. Enriquez, C. Torossian, Drinfeld associators, braid groups and explicit solutions of the Kashiwara–Vergne equations.
Publ. Math. Inst. Hautes Études Sci. 112 (2010), 143–189.
B. Enriquez, On the Drinfeld generators of $\mathfrak{grt}_1(\mathbf{k})$ and $\Gamma$-functions for associators.
Math. Res. Lett. 3 (2006), no. 2-3, 231–243.
G. Massuyeau, Formal descriptions of Turaev's loop operations. Preprint (2015), arXiv:1511.03974.
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