# Augmentations, Fillings, and Clusters

@article{Gao2020AugmentationsFA, title={Augmentations, Fillings, and Clusters}, author={Honghao Gao and Li-Chien Shen and Daping Weng}, journal={arXiv: Symplectic Geometry}, year={2020} }

We prove that the augmentation variety of any positive braid Legendrian link carries a natural cluster $\mathrm{K}_2$ structure. We present an algorithm to calculate the cluster seeds that correspond to the admissible Lagrangian fillings of the positive braid Legendrian links. Utilizing augmentations and cluster algebras, we develop a new framework to distinguish exact Lagrangian fillings.

## 14 Citations

### Non-fillable Augmentations of Twist Knots

- MathematicsInternational Mathematics Research Notices
- 2021

We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel –Ekholm–Dimitroglou Rizell…

### Lagrangian cobordism of positroid links

- Mathematics
- 2023

Casals-Gorsky-Gorsky-Simental realized all positroid strata of the complex Grassmannian as augmentation varieties of Legendrians called positroid links. We prove that the partial order on strata…

### Positive Braid Links with Infinitely Many Fillings

- Mathematics
- 2020

We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.

### ON TOPOLOGICALLY DISTINCT INFINITE FAMILIES OF EXACT LAGRANGIAN FILLINGS

- Mathematics
- 2022

. In this note we construct examples of closed connected Legendrian submanifolds in high dimensional contact vector space that admit an arbitrary finite number of topologically distinct infinite…

### DT Transformation and Infinitely Many Fillings

- Mathematics
- 2020

We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.

### Augmented Legendrian cobordism in $J^1S^1$

- Mathematics
- 2021

We consider Legendrian links and tangles in JS and J[0, 1] equipped with Morse complex families over a field F and classify them up to Legendrian cobordism. When the coefficient field is F2 this…

### Algebraic Weaves and Braid Varieties

- Mathematics
- 2020

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on…

### A note on the infinite number of exact Lagrangian fillings for spherical spuns

- MathematicsPacific Journal of Mathematics
- 2022

. In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an inﬁnite number of exact Lagrangian ﬁllings up to Hamiltonian…

### LAGRANGIAN SKELETA AND PLANE CURVE SINGULARITIES

- Mathematics
- 2020

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs (C,Λ) and Weinstein 4-manifolds W…

### Infinitely many planar fillings and symplectic Milnor fibers

- Mathematics
- 2022

We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of…

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We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.

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