Bordered Chekanov-Eliashberg DGA

• Bordered Chekanov-Eliashberg DGA is a decomposition framework that splits a Legendrian object into pieces with associated local DGAs and glues them back together via categorical pushouts.
• It employs type A, type D, and interval algebras to encode crossing data from vertical cuts in front projections, establishing a van Kampen theorem for Legendrian contact algebras.
• The method extends to higher-dimensional settings and has practical implications for tangle replacement, Mayer-Vietoris sequences, and augmentation theory in contact topology.

The bordered Chekanov-Eliashberg differential graded algebra is a cut-and-paste refinement of Legendrian contact homology in which a Legendrian knot, tangle, or more general Legendrian object is decomposed along a boundary, and differential graded algebras assigned to the pieces are glued back together by a categorical pushout. In the classical knot-theoretic setting, a front projection in standard contact $\mathbb{R}^3$ is cut by vertical lines, producing type $A$, type $D$, and interval algebras whose amalgamation recovers the Chekanov-Eliashberg algebra of the whole knot. In higher-dimensional jet spaces $J^1(M)$, an analogous construction uses holomorphic disks and, in the tractable dimensional range, gradient flow trees, yielding a van Kampen theorem for Legendrian contact algebras and associated Mayer-Vietoris and augmentation-theoretic consequences (Sivek, 2010, Harper et al., 2012).

1. Classical Legendrian contact algebra and the bordered viewpoint

The underlying invariant is the Chekanov-Eliashberg DGA. For a Legendrian knot in standard contact $\mathbb{R}^3$, it is generated combinatorially by crossings and right cusps of the front projection, and its differential counts admissible immersed disks in the front diagram (Sivek, 2010). In the more general setting of a Legendrian submanifold $L \subset J^1(N)=T^*N\times \mathbb{R}_z$, the algebra $A(L)$ is generated by Reeb chords, and the differential counts rigid $J$-holomorphic disks with boundary on $L$ (Harper et al., 2012).

The bordered construction replaces the monolithic DGA of a whole Legendrian by local DGAs attached to pieces obtained by cutting the front. The point is not merely to truncate the diagram, but to retain the algebraic effect of holomorphic or combinatorial disks that cross the cut. In the knot case this crossing data is encoded by an auxiliary interval algebra; in the higher-dimensional version it is encoded by a sub-DGA generated by the new small-action Reeb chords introduced near the dividing hypersurface (Sivek, 2010, Harper et al., 2012).

A recurrent misconception is to treat “bordered” as a bookkeeping device for endpoints alone. In the cited constructions, the border is itself algebraically active: it carries its own DGA, and the reconstruction of the global algebra is a universal colimit statement rather than a heuristic gluing rule (Sivek, 2010).

2. Type $A$, type $A$0, and interval algebras for a front cut in $A$1

Given a front diagram $A$2 of a Legendrian knot in $A$3 and a vertical dividing line meeting the diagram in $A$4 points, one cuts $A$5 into a left half $A$6 and a right half $A$7. Sivek assigns to these pieces a type $A$8 algebra $A$9 and a type $D$0 algebra $D$1, together with an auxiliary interval algebra $D$2 that records how disks cross the dividing line (Sivek, 2010).

The type $D$3 algebra $D$4 is a semi-free DGA generated by the vertices of $D$5—that is, crossings and right cusps—with the same disk-counting differential as in the full Chekanov-Eliashberg algebra. The type $D$6 algebra $D$7 is generated by the vertices of $D$8 together with free generators $D$9, $J^1(M)$0, corresponding to places where disks can cross the dividing line. For a vertex $J^1(M)$1 of $J^1(M)$2, its differential has the form

$J^1(M)$3

where the second sum runs over right half-disks starting at $J^1(M)$4 and ending on the dividing line (Sivek, 2010).

The interval algebra $J^1(M)$5 is defined independently of the knot. It is generated by the same $J^1(M)$6 and has differential

$J^1(M)$7

This algebra functions as the gluing object for disks that cross the cut (Sivek, 2010).

Piece of the cut	DGA	Role
Left half $J^1(M)$8	$J^1(M)$9	Generated by vertices of the left piece
Right half $\mathbb{R}^3$0	$\mathbb{R}^3$1	Generated by vertices and $\mathbb{R}^3$2
Dividing line with $\mathbb{R}^3$3 points	$\mathbb{R}^3$4	Encodes crossing-the-cut data

With multiple vertical cuts, the construction extends to type $\mathbb{R}^3$5 algebras for intermediate regions. A region with left and right boundaries cut out along vertical lines carries a $\mathbb{R}^3$6 generated freely by the region’s vertices and generators for the left dividing line, with canonical chain maps from the interval algebras associated to the two boundaries (Sivek, 2010).

3. Van Kampen theorem, pushouts, and tangle replacement

The central categorical statement in the knot case is that the Chekanov-Eliashberg algebra of the whole knot is the pushout of the type $\mathbb{R}^3$7 and type $\mathbb{R}^3$8 algebras over the interval algebra. Writing $\mathbb{R}^3$9 for the full DGA, the theorem states

$L \subset J^1(N)=T^*N\times \mathbb{R}_z$0

in the category of DGAs (Sivek, 2010).

The associated commutative square has maps induced by half-disk counts on the left and by the identity on the right-side generators together with the identification of $L \subset J^1(N)=T^*N\times \mathbb{R}_z$1 with their images in the full algebra. Its proof is organized around the universal property of the pushout and a combinatorial gluing analysis of admissible disks. The generators of $L \subset J^1(N)=T^*N\times \mathbb{R}_z$2 are partitioned between the two halves, while the interval algebra accounts for monomials arising from disks that traverse the dividing line (Sivek, 2010).

This pushout formalism extends recursively to finite decompositions by several vertical lines. It also descends to the characteristic algebra

$L \subset J^1(N)=T^*N\times \mathbb{R}_z$3

so the characteristic algebra of the whole knot is likewise a pushout of the characteristic algebras of the pieces. After abelianization, the corresponding pushouts become tensor products over the boundary interval algebra (Sivek, 2010).

A major application is tangle replacement. If tangles $L \subset J^1(N)=T^*N\times \mathbb{R}_z$4 and $L \subset J^1(N)=T^*N\times \mathbb{R}_z$5 have type $L \subset J^1(N)=T^*N\times \mathbb{R}_z$6 algebras related by a DGA map that is the identity on the boundary generators $L \subset J^1(N)=T^*N\times \mathbb{R}_z$7, then for any knot containing $L \subset J^1(N)=T^*N\times \mathbb{R}_z$8 and its replacement containing $L \subset J^1(N)=T^*N\times \mathbb{R}_z$9, the pushout construction yields an induced DGA map between the full knot DGAs. One consequence stated explicitly is: if $A(L)$0 admits an augmentation, so does $A(L)$1 (Sivek, 2010).

4. Higher-dimensional bordered Legendrian contact algebra in $A(L)$2

The higher-dimensional generalization replaces front cuts in $A(L)$3 by decompositions of a Legendrian submanifold $A(L)$4 along a hypersurface in the front. The ambient contact manifold is standard contact $A(L)$5, and the Legendrian contact algebra $A(L)$6 is generated by Reeb chords. The grading of a chord $A(L)$7 is

$A(L)$8

where $A(L)$9 and $J$0 count cusp crossings and $J$1 is a capping path between the endpoints of the Reeb chord (Harper et al., 2012).

The differential is defined by counts of rigid $J$2-holomorphic disks. For generic $J$3, the relevant moduli spaces have expected dimension $J$4. In the dimensional range $J$5, and for some higher-dimensional Legendrians satisfying specific front singularity conditions, the holomorphic disk counts are in one-to-one correspondence with rigid gradient flow trees, following the theory developed by Ekholm. This correspondence is the principal device that makes the bordered decomposition tractable (Harper et al., 2012).

The decomposition proceeds by a controlled Legendrian isotopy near the dividing hypersurface. One deforms $J$6 to $J$7 so that the two pieces are slightly thickened and intersect near the hypersurface. This introduces new Reeb chords of small action near the dividing set, while leaving the rest of the DGA unchanged up to quasi-isomorphism. From this one obtains three sub-DGAs:

• $J$8, generated by the Reeb chords of $J$9.
• $L$0, generated by the Reeb chords of $L$1.
• $L$2, generated by the Reeb chords of $L$3.

All three are sub-DGAs of $L$4, and $L$5 is a sub-DGA of both $L$6 and $L$7 (Harper et al., 2012).

The key analytical input is the action filtration: by Stokes’ theorem, the differential decreases action. Using this filtration together with the gradient flow tree description, one shows that the differential in $L$8 does not involve generators from $L$9 outside $A$0, and symmetrically for $A$1. The inclusion maps therefore form a commutative square

$A$2

which is a pushout square in the category of DGAs. A 3-piece version $A$3 is also established (Harper et al., 2012).

This generalizes Sivek’s bordered algebra from knots in $A$4 to Legendrian surfaces and some higher-dimensional Legendrians. It is rigorous for $A$5 and for certain higher-dimensional examples satisfying the stated front-singularity conditions (Harper et al., 2012).

5. Homological, augmentation-theoretic, and polynomial consequences

The pushout formalism induces Mayer-Vietoris sequences for linearized contact homology. In the knot case, if $A$6 is an augmentation of $A$7, with induced augmentations on $A$8 and $A$9, the linearized complexes fit into a short exact sequence and hence yield the long exact sequence

$A$00

(Sivek, 2010).

In the higher-dimensional setting, the pushout square similarly induces a Mayer-Vietoris sequence for linearized contact homology for all choices of augmentations: $A$01 (Harper et al., 2012).

Augmentation varieties also behave well under bordered decompositions. For the connect sum $A$02, the augmentation variety introduced by Ng satisfies the direct product formula

$A$03

The same paper states an analogous relation, with shifts, for Poincaré polynomials of linearized contact homology (Harper et al., 2012).

In the knot case, the bordered machinery is also used to compute linearized invariants of Legendrian Whitehead doubles. If $A$04, then for the Legendrian Whitehead double $A$05 one obtains

$A$06

The same method is used to prove that Whitehead doubles of prime knots can have arbitrarily many Chekanov polynomials (Sivek, 2010).

These consequences clarify the role of bordered structures: they are not only decomposition theorems for DGAs, but mechanisms for transporting augmentation data, linearized homology, and characteristic algebra information across local modifications (Sivek, 2010, Harper et al., 2012).

6. Generalizations to graphs, singular Legendrians, and rational SFT

The bordered formalism has been extended in several directions. For bordered Legendrian graphs, one works with a diagram of DGAs

$A$07

where the border DGAs are modeled as upper-triangular matrix DGAs depending on the boundary combinatorics and Maslov potentials. Concatenation of bordered graphs corresponds to pushout diagrams, and the augmentation number over a finite field is shown to agree with a ruling polynomial defined by normal-ruling combinatorics (An et al., 2019).

A related construction for Legendrian graphs and tangles in standard contact $A$08 introduces countably many generators at each vertex and a generalized stable-tame equivalence. Its van Kampen-type theorem describes tangle replacement by a pushout square

$A$09

and recovers known algebraic constructions for Legendrian links by suitable operations at the vertices (An et al., 2018).

For singular Legendrians, particularly skeleta of Weinstein manifolds, the Chekanov-Eliashberg dg-algebra is extended to an object

$A$10

with a push-out diagram expressing cut-and-paste along a shared Weinstein subdomain. This framework is used to give direct proofs of wrapped Floer cohomology push-out diagrams and to establish a quasi-isomorphism with Chekanov-Eliashberg dg-algebras having coefficients in chains on the based loop space (Asplund et al., 2021).

A further extension appears in bordered Legendrian rational symplectic field theory. There, one assigns commutative DGAs $A$11, $A$12, and $A$13 to the left half, right half, and middle boundary data of a split Legendrian knot, and the differential incorporates disks with multiple positive punctures together with string topology terms. The resulting square

$A$14

is a pushout of commutative DGAs. This construction explicitly extends the bordered Chekanov-Eliashberg DGA beyond the one-positive-puncture regime (Wlodek, 9 Sep 2025).

In a broader algebraic context, the decomposition methods of cornered Floer homology assign a differential graded 2-algebra to the circle and algebra-modules to surfaces with boundary, with natural gluing properties. This suggests an algebraic template for codimension-two boundary data in bordered Legendrian theories, although the bordered Chekanov-Eliashberg constructions themselves are formulated directly in DGA terms rather than through sequential 2-algebras (Douglas et al., 2011).

The cumulative picture is that the bordered Chekanov-Eliashberg DGA is both a specific invariant and a general methodology. Its basic theorem is a pushout statement for contact-topological DGAs, and its later extensions preserve the same local-to-global principle while enlarging the class of Legendrian objects and the holomorphic-curve phenomena encoded algebraically (Sivek, 2010, Harper et al., 2012).