Lagrangian Fillable Legendrian Knots

• Lagrangian fillable Legendrian knots are submanifolds in contact 3-manifolds that bound Lagrangian surfaces, offering a bridge between contact topology and symplectic geometry.
• Analysis employs DG-algebraic invariants, combinatorial grid diagrams, and positivity criteria to classify fillability and construct explicit fillings via handle attachments and pseudoholomorphic methods.
• Obstructions to fillability arise from augmentation varieties and torsion in linearized homology, while concordance studies reveal non-regular phenomena and multiple filling classifications.

A Lagrangian fillable Legendrian knot is a Legendrian submanifold in a contact 3-manifold, typically the standard contact 3-sphere, which bounds an exact Lagrangian surface in a symplectic filling such as the standard symplectic 4-ball. The analysis of these objects lies at the intersection of contact topology, symplectic geometry, and low-dimensional topology, and features a deep interplay between combinatorial, geometric, and algebraic invariants. This article provides a technical overview focusing on the classification, construction, and obstructions to Lagrangian filling for Legendrian knots, emphasizing the role of DG-algebraic invariants, combinatorial moves, and concordance phenomena.

1. Algebraic Invariants and the DGA Framework

Legendrian knots in both $\mathbb{R}^{3}$ and more general 3-manifolds (including lens spaces %%%%1%%%%) admit a Chekanov–Eliashberg differential graded algebra (DGA) invariant, which encodes the counts of rigid pseudo-holomorphic disks with boundaries on the shifted Legendrian in the symplectization or 1-jet space. For knots in tight lens spaces, grid number one diagrams provide an efficient encoding for Legendrian knot types and lead to explicit constructions of the DGA. In such diagrams for $L(p,q)$, two distinguished basepoints determine the data of the knot and, after special projection and resolution, yield a Lagrangian projection %%%%3%%%% on which the DGA is based.

Key features of this framework:

• Generators: The DGA is generated (over $\mathbb{Z}_2$ or a group ring) by symbols associated to Reeb chords—namely pairs $a_i, b_i$ per crossing in $\Gamma$.
• Differential: Defined combinatorially via counts over “admissible discs” $f: (D^2, \partial D^2) \to (S^2, \Gamma)$ whose boundaries encode the combinatorial structure of grid diagrams and the length data of chords. The disc defect formula

$$n(R) = -a(R) + \sum_{\text{corners }} \epsilon(i) \ell(x_i)$$

controls which discs contribute to the DGA differential.

• Grading: Determined using capping paths, rotation and winding numbers, with explicit formulas such as $$|a| = 2\lceil r(\eta)\rceil - 2((p-1)/p)w_N(\eta) - 1 + 4n(D_\eta)$$, and $|b| = 3 - |a|$.

Augmentations are key algebraic tools: a graded algebra homomorphism $\epsilon : \mathcal{A}(K) \to \mathbb{Z}_2$ with $\epsilon \circ \partial = 0$ and $\epsilon(x)=0$ for $|x|\ne0$. Existence of augmentations is often taken as indicative—though not sufficient in general—of Lagrangian fillability, as they often arise from (or are induced by) Lagrangian surface fillings.

2. Combinatorial and Fillability Criteria

The relationship between positivity and fillability was sharply illuminated by the result that all positive knots (those that admit diagrams with only positive crossings) are Lagrangian fillable; more precisely, every positive knot has a Legendrian representative that admits an exact, embedded Lagrangian filling. This encompasses an inductive construction using oriented normal rulings where all crossings are switched, yielding a decomposable filling via 0- and 1-handle attachments (Hayden et al., 2013).

The hierarchy of positivity is summarized via inclusions:

$$\text{Braid Positive} \subset \text{Positive} \subset \text{Strongly Quasi-Positive} \subset \text{Quasi-Positive}.$$

While every Lagrangian fillable Legendrian knot is quasi-positive (results of Eliashberg, Boileau, Orevkov), the converse does not hold: strong quasi-positivity is independent of fillability, with explicit construction of quasi-positive, non-fillable and fillable, non-strongly-quasi-positive examples.

For Legendrian 4-plat knots, an explicit combinatorial characterization shows that fillability is equivalent to each negative band having at most two crossings, and each internal band (i.e., not the first or last) having at least two (Lipman et al., 2017). Additionally, any fillable Legendrian 4-plat knot has a smooth type that is positive, confirming a tight correlation between diagrammatic positivity and Lagrangian filling in this class.

3. Unique and Multiple Lagrangian Fillings: Classification & Invariants

The classification of fillings can display both uniqueness and profusion depending on the knot class:

• The standard Legendrian Hopf link in $(S^3, \xi_{std})$ has exactly two embedded exact Lagrangian fillings up to compactly supported Hamiltonian isotopy, corresponding to different winding numbers in the Lefschetz fibration base (Thomson, 18 Jun 2025).
• In contrast, the standard Legendrian $(2,n)$ torus knot or link with maximal Thurston–Bennequin number allows exactly $C_n$ non–isotopic exact Lagrangian fillings, with $C_n$ the $n^{\text{th}}$ Catalan number (Pan, 2016). The distinctness is witnessed via combinatorial invariants extracted from augmentations induced by the filling, as the structure of the induced augmentation distinguishes fillings up to isotopy.

Fillings of the same boundary can be smoothly non-isotopic or have non-homeomorphic exteriors. This arises in the construction of Lagrangian ribbon disks for mirror knots such as $\overline{9_{46}}$, where different choices in the sequence of pinch moves result in distinct fillings; handlebody decompositions and invariants such as the fundamental group or adjunction inequality detect non-equivalence (Li et al., 2019).

4. Augmentation Varieties and Obstructions

The structure of the augmentation variety—the space of augmentations of the DGA—encodes deep information about fillability:

• A Floer-theoretic argument shows that the set of fillable augmentations of a Legendrian knot must lie in the injective image of an algebraic torus, with dimension equal to the first Betti number $b_1$ of the filling. For certain Legendrian twist knots, explicit examples of augmentations exist (labelled $\epsilon_3$) that do not lie in such subvariety and hence cannot be filled by any orientable Lagrangian surface; others are geometric and arise from explicit fillings (Gao et al., 2021).
• The linearized homology for augmentations can contain torsion over $\mathbb{Z}$ even when their mod 2 reductions are geometric and come from actual Lagrangian fillings, giving rise to augmentations that are not realized by any embedded filling (Lipshitz et al., 2023). For instance, specific Legendrian representatives of $m(8_{21})$ admit augmentations with $Z/n$ torsion in linearized homology incompatible with the Seidel isomorphism if a filling existed, providing an obstruction to geometric realizability.

Key formulas: $tb(\Lambda) = -\chi(L)$ for orientable fillings and

$tb(\Lambda) = -\chi(L) - e(L)$

if the filling is non-orientable, $e(L)$ being the normal Euler number.

5. Construction Methods and Symplectic Techniques

Techniques for constructing fillings include:

• Decomposable fillings: Built from sequences of handle attachments (e.g., 0- and 1-handles) corresponding to moves in the front projection (stabilization, pinch moves, etc.) (Hayden et al., 2013, Conway et al., 2017). For any Legendrian, stabilization produces a knot cobordant to the standard unknot, which admits explicit Lagrangian caps (Lin, 2013).
• Cluster algebra and sheaf-theoretic methods: Certain fillings correspond to cluster charts or 'seeds' in the moduli of local systems or constructible sheaves microsupported in the knot; isotopy between fillings is distinguished via cluster transformations (Shende et al., 2015). For alternating/plabic front diagrams, the number of fillings can be identified with combinatorial numbers (e.g., Catalan).
• Neck-stretching and pseudoholomorphic techniques: For classification, neck-stretching and SFT compactness, combined with Fredholm index calculations and positivity of intersection, are key for control of degenerations and construction of isotopies (Thomson, 18 Jun 2025).

6. Concordances, Cobordism, and Non-Regular Phenomena

Lagrangian concordance between Legendrian knots is a rich topic. While decomposable concordances (built from elementary moves) induce partial orders on Legendrian knots with certain properties (e.g., strongly homotopy-ribbon), the existence of non-regular, non-decomposable Lagrangian concordances can violate antisymmetry. For every decomposable Lagrangian concordance $L$ from $\Lambda_-$ to $\Lambda_+$, there exists a non-regular concordance from a stabilized Legendrian satellite of $\Lambda_+$ to one of $\Lambda_-$ (produced using Legendrian Whitehead doubles and inversion of totally real concordances via an $h$-principle), resulting in pairs of fillable knots with concordances in both directions that cannot both be regular (Rizell et al., 16 Sep 2025). This indicates that the Lagrangian concordance relation on fillable knots is not a partial order.

Additionally, non-orientable exact Lagrangian endocobordisms exist for all stabilized Legendrian knots but are obstructed for exactly fillable knots (by the Seidel isomorphism and augmentation finiteness) (Capovilla-Searle et al., 2015).

7. Non-orientable Fillings and Rigidity Phenomena

Non-orientable exact Lagrangian fillings, while common in the smooth category, are highly restricted in the symplectic (exact, decomposable) one. For alternating and plus-adequate knots, a canonical ruling on the front diagram provides both combinatorial obstructions (via e.g. the mod 2 resolution linking number) and constructive tools for non-orientable fillings (Chen et al., 2022). The set of normal Euler numbers and crosscap numbers for such fillings is finite and minimal among all smooth non-orientable fillings, in stark contrast to the unbounded possibilities in the smooth category.