Related papers: Cluster varieties from Legendrian knots

We construct exact Lagrangian fillings of Legendrian torus links $\Lambda(k, n-k)$ that are fixed by a Legendrian loop that acts by $2\pi\ell/n$ rotation. Using these rotationally symmetric fillings, we produce fillings of the corresponding…

In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs.…

We give the full classification of smooth toric Legendrian subvarieties in projective space. We also prove that under some minor assumptions the group of linear automorphisms preserving given Legendrian subvariety preserves the contact…

This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…

Let $X$ be a K3 surface and let $\text{Spl}(r;c_1,c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. Under suitable assumptions, to a pair $(F,W)$ (respectively, $(F,V)$) where $F\in…

We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and…

We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov's plabic graphs. This description is a natural generalization of the description given…

By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such…

We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the…

We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on…

This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the…

We investigate when a Legendrian knot in standard contact $\mathbb{R}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability,…

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…

This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables…

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are…

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…

This paper presents an algorithm to deform any Legendrian singularity to a nearby Legendrian subvariety with singularities of a simple combinatorial nature. Furthermore, the category of microlocal sheaves on the original Legendrian…

We prove that Legendrian and transverse links in overtwisted contact structures having overtwisted complements can be classified coarsely by their classical invariants. We further prove that any coarse equivalence class of loose links has…

In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing…

Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and…