Splicing braid varieties

TL;DR

This paper introduces a new open cover for braid varieties associated with positive braids, explores their cluster structures, and proves conjectures in key special cases, advancing understanding of their algebraic and geometric properties.

Contribution

It defines a family of open sets in braid varieties, conjectures their cluster structure, and proves these conjectures in important special cases.

Findings

Open sets form an open cover of braid varieties.

Each open set is isomorphic to a product of two braid varieties.

Conjectures about cluster structures are proved in special cases.

Abstract

For a positive braid $\beta \in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(\beta)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(\beta)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater than the length of $\beta$. For fixed $r$, the sets $\mathcal{U}_{r, w}$ form an open cover of $X(\beta)$. We conjecture that $\mathcal{U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb{C}[X(\beta)]$ and that $\mathcal{U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal{U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures.