Splicing braid varieties
Eugene Gorsky, Soyeon Kim, Tonie Scroggin, Jos\'e Simental

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 , we consider the braid variety . We define a family of open sets in , where is a permutation and is a positive integer no greater than the length of . For fixed , the sets form an open cover of . We conjecture that is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on and that admits a cluster structure given by freezing these variables. Moreover, we show that 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.
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