Subsection 7.2.4. (0PH7) · ScalingStacks

7.2.4. Chord diagrams as singular curves

We describe here the relation between singular curves and chord (or arc) diagrams.

We define a chord diagram to be to be a triple $({\mathcal{Z}},{\mathbf{a}})$ where

•

${\mathcal{Z}}$ is a closed oriented $1$ -dimensional manifold (i.e., a finite disjoint union of copies of $S^{1}$ and $[0,1]$ )
•

${\mathbf{a}}$ is a finite set of pairs of points of $\mathring{{\mathcal{Z}}}$ , all of which are distinct.

A chord diagram gives rise to a smooth oriented curve $\tilde{Z}=\mathring{{\mathcal{Z}}}$ with the following relation: given $z\neq z^{\prime}$ , we have $z\sim z^{\prime}$ if $\{z,z^{\prime}\}\in{\mathbf{a}}$ . We obtain an oriented curve $Z=\tilde{Z}/\!\sim$ and a map $\mu:\bigcup_{\{z,z^{\prime}\}\in{\mathbf{a}}}\{z,z^{\prime}\}\to Z_{exc}$ inducing a bijection ${\mathbf{a}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z_{exc}$ .

Up to suitable isomorphism, this defines a bijection from chord diagrams to oriented singular curves with $n_{z}\in\{2,4\}$ for all $z$ .

0P9K

Convention 7.2.12. We will use the above bijection composed with the reversal of all orientations when identifying chord diagrams with certain singular curves. This orientation reversal is related to the usual direction reversal between arrows in a quiver and morphisms in the corresponding path category, and to the time-reversal of graphs mentioned in Example 7.3.8 below.

When ${\mathcal{Z}}$ is a union of intervals, we recover the notion of (possibly degenerate) arc diagram due to Zarev [Za, Definition 2.1] (compare Example 7.2.11 and [Za, Figures 3 and 4]).

The chord diagrams such that the singular curve $Z$ is connected and $k>0$ correspond to the chord diagrams of [AnChePeReiSu].

Zarev’s definition generalizes that of pointed matched circles due to Lipshitz, Ozsváth and Thurston [LiOzTh1, §3.2]: they correspond to the case where ${\mathcal{Z}}$ is a single interval ( $\mathring{{\mathcal{Z}}}$ is obtained from the circle considered in [LiOzTh1] by removing its basepoint).

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2