Related papers: Augmentations and immersed Lagrangian fillings

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such…

Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times \Lambda$ be a cylinder over a Legendrian submanifold in the contactisation. We show…

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $\Lambda_-$ to $\Lambda_+$, we construct a functor $\Phi_L^*: Sh^c_{\Lambda_+}(M) \rightarrow Sh^c_{\Lambda_-}(M) \otimes_{C_{-*}(\Omega_*\Lambda_-)} C_{-*}(\Omega_*L)$…

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with certain symmetries as seeds…

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many…

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 isotopy. They are obtained from the examples…

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…

A connection between holomorphic and generating family invariants of Legendrian knots is established; namely, that the existence of a ruling (or decomposition) of a Legendrian knot is equivalent to the existence of an augmentation of its…

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces…

In this note we construct augmentations of Chekanov-Eliashberg algebras of certain high dimensional Legendrian submanifolds that are not induced by exact Lagrangian fillings. The obstructions to the existence of exact Lagrangian fillings…

Given a Legendrian submanifold in any dimension, we prove that two augmentations are isomorphic within the positive augmentation category exactly when they differ by a combination of a dga homotopy and a dilation. This extends the…

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type $\mathsf{ADE}$ or affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with…

This is the third in a series 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 prove that for connected…

For an exact Lagrangian cobordism $L$ between Legendrians in $J^1(M)$ from $\Lambda_-$ to $\Lambda_+$ whose Legendrian lift is $\widetilde{L}$, we prove that sheaves in $Sh_{\widetilde{L}}(M \times \mathbb{R} \times \mathbb{R}_{>0})$ are…

We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian $2$-bridge links is…

We study augmentations of a Legendrian surface $L$ in the $1$-jet space, $J^1M$, of a surface $M$. We introduce two types of algebraic/combinatorial structures related to the front projection of $L$ that we call chain homotopy diagrams…

Let $\Lambda$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $\Lambda$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $\Lambda$, and such…

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of…

Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links…

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{\infty}$-category, which lifts the set of…