We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K₂ (aka. \(\mathcal{A}\)-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in Ekholm et al. (J. Eur. Math. Soc. 18(11):2627–2689, 2016), we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

我们通过 Floer 理论方法研究正辫状传奇链接，并证明它们的增强变体是簇 K ₂（又名\(\mathcal{A}\) -）变体。使用 Ekholm 等人中的 Legendrian 链接的精确拉格朗日配边法。 （J. Eur. Math. Soc. 18(11):2627–2689, 2016），我们证明了正编织勒让德链接的一大类精确拉格朗日填充与其增强变种的簇种子相对应。我们解决了正编织勒让德链接的无限填充问题；即，只要正辫状勒让德链接不是 ADE 类型，它就允许无限多个精确的拉格朗日填充，直到哈密顿同位素。