Chekanov–Eliashberg DGA Overview

• Chekanov–Eliashberg DGA is a holomorphic curve invariant for Legendrian knots that encodes combinatorial data using Reeb chords and cusp generators.
• It is constructed with generators from crossings and cusps whose differential counts rigid J-holomorphic disks, ensuring the key property that ∂² = 0.
• Extensions of the DGA to singular Legendrians and graphs, along with links to Fukaya categories, drive advances in contact and symplectic topology.

The Chekanov–Eliashberg Differential Graded Algebra (DGA) is a holomorphic curve invariant in contact and symplectic topology, originally defined for Legendrian knots in $\mathbb{R}^3$ and now extended to higher dimensions, singular settings, and more elaborate algebraic structures. The DGA encodes combinatorial and holomorphic data of Legendrian submanifolds via generators (corresponding to Reeb chords and crossings) and a differential determined by counts of rigid $J$-holomorphic curves. This algebraic invariant is a central tool for distinguishing Legendrian isotopy classes, computing contact homology, and building bridges to Floer-theoretic and representation-theoretic categories.

1. Definitions and Construction of the Chekanov–Eliashberg DGA

For a Legendrian knot or link $\Lambda$ in the standard contact $(\mathbb{R}^3, \ker(dz-ydx))$, the Chekanov–Eliashberg DGA $(A, \partial)$ is generated by crossings and right cusps of the front projection of $\Lambda$; left cusps contribute to the Maslov potential used for grading but are not generators. The algebra is typically defined over a ring such as $\mathbb{Z}[t, t^{-1}]$, or a field $F$, where $t$ encodes basepoint and rotation information.

• Generators: Crossings ($c$), right cusps ($q$), and sometimes extra augmented crossings from a "dipping" procedure.
• Grading: Assignments via a Maslov potential $\mu$: $|q| = 1$ for a right cusp, $|c| = \mu(T) - \mu(B)$ for a crossing, with $T$ and $B$ the top and bottom strands.
• Differential:

$$\partial c = \sum_{f \in M(c)_{(1)}} \text{sgn}(f) \cdot \text{monomial}(f)$$

where $M(c)_{(1)}$ denotes moduli of admissible disks contributing to the differential of $c$.

The crucial property is $\partial^2 = 0$, proved via analytic arguments involving coherent orientations in moduli spaces of disks.

2. Augmentations, Linearization, and Rulings

An augmentation is a DGA map $\epsilon: A \to F$ (for a field $F$) satisfying $\epsilon(1)=1$ and $\epsilon \circ \partial = 0$. Augmentations are parametrized by rulings of the front diagram, which are combinatorial partitions of the strands. There is an explicit correspondence between augmentations and normal rulings:

• Existence of an augmentation $\iff$ existence of a ruling of the front (Leverson, 2014).
• Even graded augmentations must send $t$ to $-1$; for odd graded cases, augmentation values satisfy $\epsilon(t) = -x^2$, $x \in F^*$.

Property (R) requires that augmentation values on "dip generators" encode the strand pairing in the ruling.

3. Pushouts, Locality, and Tangle Replacement

Legendrian invariants from the Chekanov–Eliashberg DGA display locality properties. Cutting a front diagram along vertical lines and assigning DGAs to each piece allow global invariants to be recovered as the pushout of local DGAs (Sivek, 2010). Formally, for a decomposition

$$I_n \rightarrow D(K^D) \ \downarrow \,\qquad \downarrow \ A(K^A) \rightarrow Ch(K)$$

the Chekanov–Eliashberg invariant of $K$ is the pushout in the category of DGAs.

Compatible tangle replacements (including moves such as "parallel break," "unhooking a clasp," or S/Z tangle swaps) induce morphisms of DGAs via pushouts, ensuring invariants such as augmentations, Chekanov polynomials, and characteristic algebras are stably related under such replacements.

4. Computation and Homology Invariants

Given an augmentation, the Chekanov–Eliashberg DGA can be linearized, yielding a finite-dimensional chain complex whose homology computes Legendrian contact homology invariants (Chekanov polynomials). The polynomial

$P_\epsilon(t) = \sum h_i t^i$

encodes dimensions $h_i$ of homology groups in degree $i$ (Casey et al., 2014). Uniqueness results, such as the property that a front with exactly four cusps admits at most one Chekanov polynomial, follow from detailed combinatorial analysis (Morse complex sequences, chord path counts) in the linearized setting.

5. Characteristic Algebra, Representations, and Arnold-type Bounds

The characteristic algebra $C_\Lambda$—the quotient of $A(\Lambda)$ by the ideal $\langle \partial(A) \rangle$—admits finite-dimensional representations, which generalize augmentations (which are 1-dimensional representations). Existence of a $k$-dimensional representation yields Arnold-type lower bounds for the number of Reeb chords in terms of Betti numbers and chord gradings (Rizell et al., 2014). In cases where no augmentation exists, representations of the characteristic algebra still provide rigidity and lower bounds for chord counts.

6. Extensions to Singular Legendrians, Graphs, and SFT-type Deformations

Recent developments have generalized the Chekanov–Eliashberg DGA to:

• Singular Legendrians and skeleta of Weinstein domains: The invariant is constructed from the Legendrian attaching spheres and is formally defined as $CE^*((V, h); W) := CE^*(\Sigma(h); W_V^0)$, capturing top and subcritical handle data and allowing for pushout diagrams reflecting surgery and gluing formulas (Asplund et al., 2021).
• Legendrian graphs: The DGA includes countably many generators at each vertex and peripheral structures for invariance under vertex moves and tangle replacements. Pushout theorems analogous to van Kampen hold for the DGAs assigned to graphs and tangles (An et al., 2018).
• Second-order DGA and SFT-type deformations: Beyond the classical DGA, algebraic structures that encode failure of strict derivation (via antibrackets and $\hbar$ extensions) arise; their differentials count $J$-holomorphic disks and annuli, with invariance established through careful sign and orientation analysis (Dukic, 9 Sep 2024).

7. Representation Theory and Fukaya Categories

Finite-dimensional representations of the Chekanov–Eliashberg DGA, constructed from closed exact Lagrangian submanifolds in a Weinstein manifold, directly connect the compact Fukaya category of the ambient manifold to the derived category of DGA modules. The isomorphism

$$HF(L_0, L_1) \cong H^*R\hom_{\mathcal A}(V_{L_0}, V_{L_1})$$

provides a categorical bridge between symplectic and contact topology, with modules $V_L$ explicitly constructed from intersection data between $L$ and cocores of handles (Chantraine et al., 28 Aug 2025).

Table: Key Algebraic Structures

Structure	Generators	Differential Definition
CE DGA (standard)	crossings, cusps	holomorphic disks, combinatorial disk counts
Characteristic Algebra	CE DGA quotiented	counts under quotient by differential image
Graph DGA	crossings, vertices	regular & infinitesimal disk counts; peripheral maps
Second-order DGA	tensors + $\hbar$ terms	disks, annuli, quantum corrections

These developments position the Chekanov–Eliashberg DGA as a cornerstone in modern contact and symplectic topology, enabling effective computation of Legendrian invariants, sharp control under local diagram moves, and deep categorical connections to Floer theory, representation theory, and mirror symmetry.