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Splicing braid varieties
- Part of
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- Journal:
- Canadian Mathematical Communications / Volume 1 / 2026
- Published online by Cambridge University Press:
- 06 April 2026, e4
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- Article
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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 
$\mathbb {C}[X(\beta )]$ 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 
$\mathcal {U}_{r,w}$ 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.
