Augmentations, fillings, and clusters.(English)Zbl 07862380
Summary: We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster \(K_{2}\) (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.
MSC:
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 53D10 | Contact manifolds (general theory) |
| 53D12 | Lagrangian submanifolds; Maslov index |
| 13F60 | Cluster algebras |
Cite
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