Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[\pi]$ of the fundamental group $\pi$ 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 $\mathbb{Q}[\pi]$ 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 $H:=H_1(\pi;\mathbb{Q})$. 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 $\mathbb{Q}[\pi]$ and $T(H)$ that arises from a Drinfeld associator by embedding $\pi$ 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.

中文翻译：

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Turaev 循环操作的形式化描述

利用环的交和自交，图拉耶夫在 70 年代引入了对带边界曲面的基本群 $\pi$ 的代数 $\mathbb{Q}[\pi]$ 的两个基本运算。第一个运算是二元运算，测量曲面上两条定向曲线的交点，而第二个运算是一元运算，计算定向曲线的自交。已经知道，当群代数 $\mathbb{Q}[\pi]$ 就其增广理想的幂完成并且被适当地识别为张量代数的度完成时，Turaev 的交集配对具有代数描述$H:=H_1(\pi;\mathbb{Q})$的$T(H)$。在本文中，我们在具有 $p$ 穿孔的圆盘的情况下获得了 Turaev 自交图的类似代数描述。在这里，我们考虑通过将 $\pi$ 嵌入 $(p+1 )$ 股; 我们的代数描述涉及由关联子明确确定的形式幂级数。该证明基于图拉耶夫循环操作的一些三维公式，其中涉及 $2$-strand 纯辫子，并针对任何具有边界的表面显示。