Abstract For a positive braid Br^+₊, we consider the braid variety X (). We define a family of open sets Uₑ, ₖ in X (), where w Sₖ is a permutation and r is a positive integer no greater than the length of. For fixed r, the sets Uₑ, ₖ form an open cover of X (). We conjecture that Uₑ, ₖ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on CX () constructed in Casals et al. (2025, J. Amer. Math. Soc. 38, 369–479), Galashin et al. (2026, Invent. Math. 243, 1079–1127), and Galashin et al. (2022, Braid variety cluster structures, I: 3D plabic graphs) and that Uₑ, ₖ admits a cluster structure given by freezing these variables. Moreover, we show that Uₑ, ₖ 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.
Gorsky et al. (Thu,) studied this question.