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arxiv: 2604.22053 · v2 · submitted 2026-04-23 · 🧮 math.SG · math.GT

Recognition: 2 theorem links

· Lean Theorem

Invariants of Legendrian knots in thickened convex surfaces

Nancy Mae Eagles, Zijian Rong

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:36 UTC · model grok-4.3

classification 🧮 math.SG math.GT
keywords Legendrian knotsdifferential graded algebraconvex surfacesdividing setReeb chordsChekanov-Eliashberg invariantLegendrian isotopy
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The pith

A differential graded algebra built from Reeb chords and immersed polygons bounded by the dividing set gives an isotopy invariant for Legendrian knots in thickened convex surfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a differential graded algebra for a Legendrian knot in a thickened convex surface Σ×ℝ by using the dividing set Γ on Σ. Generators are the Reeb chords of the knot, and the differential counts immersed polygons in the projection to Σ whose boundaries lie on the projected knot together with Γ. The authors prove that this differential squares to zero and that the stable tame isomorphism type of the algebra remains unchanged under Legendrian isotopy. The resulting invariant distinguishes certain Legendrian knots that cannot be separated by the classical rotation number or Thurston-Bennequin number.

Core claim

A differential graded algebra is associated to a Legendrian knot Λ in Σ×ℝ. It is generated by the Reeb chords of Λ, and its differential counts immersed polygons in the projection π(Σ×ℝ)→Σ with boundary on π(Λ)∪Γ. The differential squares to zero, and the stable tame isomorphism class of the algebra is invariant under Legendrian isotopy of Λ.

What carries the argument

The DGA generated by Reeb chords whose differential counts immersed polygons with boundary on the projected Legendrian union the dividing set Γ.

If this is right

  • The algebra supplies a computable invariant that refines the classical numerical invariants for Legendrian knots in these manifolds.
  • Examples computed in the paper show concrete pairs of knots that become distinguishable once the new algebra is evaluated.
  • The construction adapts the polygon-counting method of the Chekanov-Eliashberg DGA to surfaces equipped with dividing sets.
  • The invariance under isotopy implies that the algebra descends to a well-defined object on the isotopy class of the Legendrian knot.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same polygon-counting technique might extend to Legendrian knots in other contact 3-manifolds that admit convex surfaces.
  • The immersed polygons counted here could be interpreted as boundaries of holomorphic disks in a suitable symplectic filling of the thickened surface.
  • If the algebra can be computed algorithmically for surfaces of higher genus, it would give a practical tool for classifying Legendrian knots beyond the standard numerical invariants.

Load-bearing premise

The count of immersed polygons is finite, independent of choices, and only transverse polygons contribute.

What would settle it

An explicit Legendrian isotopy between two knots whose associated DGAs are not stably tame isomorphic would contradict the invariance claim.

Figures

Figures reproduced from arXiv: 2604.22053 by Nancy Mae Eagles, Zijian Rong.

Figure 1
Figure 1. Figure 1: Two Legendrian knots in Σ2 that are not Legendrian isotopic. The surface Σ2 is constructed by identifying the pair of pants Σ` and Σ´ along the dividing set Γ in the obvious way. Γ and several negative ends asymptotic to Reeb chords along Γ or double-points in Σ˘. The formula does not involve any J-holomorphic curves with more than one negative ends along Γ and we currently do not have any good conceptual … view at source ↗
Figure 2
Figure 2. Figure 2: Contractible Reeb orbits are forbidden. the sutured Legendrian contact homology dg-algebra AredpY,Λq is quasi-isomorphic to the combinatorially-defined DGA Ap in this paper. 1.3. Organization. This paper is organized as follows. Section 2 recalls some necessary background from contact geometry and introduces the contact manifold in which we will work. It also includes a review of some algebra background. S… view at source ↗
Figure 3
Figure 3. Figure 3: Relative orientations on Σ˘. Definition 2.1. Let A be a semi-free DGA over Z2 generated by c1, c2 . . . . An ordering cσp1q ă cσp2q ă . . . of the generators of A (where σ is a permutation) induces a filtration Z2 “ F 0A Ă F 1A Ă F 2A Ă . . . by Fi “ Z2xcσp1q , . . . , cσpiqy. An elementary automorphism of A is a DGA automorphism from A Ñ A such that there exists a generator cσpiq with cσpiq ÞÑ cσpiq ` v c… view at source ↗
Figure 5
Figure 5. Figure 5: A Legendrian unknot in T 2 ˆ R. Γ Σ` Σ´ Γ Σ` Σ´ Γ Σ` Σ´ view at source ↗
Figure 6
Figure 6. Figure 6: Another Legendrian knot in T 2 ˆ R. 3.2. Legendrian Reidemeister theorem for the convex surface projection. Definition 3.4. Λ has a good projection if πpΛq is an immersed submanifold of Σ whose only self-intersections are transverse double-points off Γ and πpΛq intersects Γ transversely. Definition 3.5. Let Λt be a smooth isotopy in pΣ ˆ R, ξΓq between two Legendrian knots Λ0 and Λ1 having good projections… view at source ↗
Figure 7
Figure 7. Figure 7: Reidemeister moves for the convex surface projection. (a) There exists an arbitrarily small Legendrian perturbation Λ 1 of Λ such that Λ 1 has a good projection. (b) There exists an arbitrarily small Legendrian perturbation Λ 1 t of Λt such that Λ 1 0 “ Λ0, Λ 1 1 “ Λ1, and Λ 1 t has a good projection. 3.3. Classical invariants. Now we compute the classical invariants of Λ in terms of its convex surface pro… view at source ↗
Figure 8
Figure 8. Figure 8: The sign of a crossing of Λ over Λ1 . Λ 1 Λ Γ Λ 1 Λ (a) (b) Λ Λ 1 view at source ↗
Figure 9
Figure 9. Figure 9: The contact framing of Λ. Remark 3.8. One can easily check that the formula for tbpΛq is invariant under Legendrian Reidemeister moves for the convex surface projection. The rotation number rotpΛq can be computed in a similar way as the grading of gener￾ators. See Section 4.3.2 for details view at source ↗
Figure 10
Figure 10. Figure 10: Orientations of corners in Σ˘. Definition 4.1 (Rigid polygon). A polygon is rigid if it has exactly one positive end, and arbitrarily many negative ends. For a Reeb chord c along Γ, denote by M˘pc, d1, . . . , dnq the moduli space of rigid polygons in Σ˘ with a positive end c in Σ˘ or on Γ and negative ends at Reeb chords d1, . . . , dn in Σ˘. As in the standard construction of [Che02, Sec. 6] for Legendr… view at source ↗
Figure 11
Figure 11. Figure 11: (a) A rigid polygon with positive end c˘ in Σ˘ and (b) a rigid polygon with positive end cΓ on Γ. Definition 4.2 (Height of a Reeb chord). Fix Σ1 ` Ă Σ` such that all double-points of πpΛq in Σ` are contained in Σ1 `. Then fix a contact form α0 “ β `udt on ΣˆR for ξΓ such that u ” 1 on Σ1 `. Let c be either a double-point of πpΛq or a (possibly self-overlapping) arc along Γ connecting two points in πpΛq X… view at source ↗
Figure 12
Figure 12. Figure 12: Height of a Reeb chord along Γ Proof. Throughout the proof we use P to denote P ´ Rpγ ∥ c q. Lift P to a polygon P in Σ 1 ` ˆ R and write ιpPq “ P. The sides of P consist of arcs γ1, . . . , γn of Λ and γc1 , . . . , γcn for some double-points c1, . . . , cn corresponding to the ends of P. Then ż tγiu n i“1Ytγcj u n j“1 α “ ÿn j“1 ż γcj α since α|γi “ 0 “ ÿ cPQ` Hpcq ´ ÿ cPQ´ Hpcq. At the same time, we ha… view at source ↗
Figure 13
Figure 13. Figure 13: Resolution of double points along a capping path γc. ∗We note that we keep the corners associated to the crossing c, although this is not required for defining |c|. Indeed, for any capping surface of γc, the contributions coming from these corners cancel. Thus, one could equivalently simplify the computation by applying the standard Seifert resolution at c, replacing the crossing with two disjoint, smooth… view at source ↗
Figure 15
Figure 15. Figure 15: The standard unknot in T 2 ˆ R. c d Γ γc Sc Sd view at source ↗
Figure 16
Figure 16. Figure 16: Capping surfaces for Reeb chords c and d. Example 4.12. Consider the standard unknot in view at source ↗
Figure 17
Figure 17. Figure 17: The standard unknot in Σ2 ˆ R. Example 4.13. Consider the example in view at source ↗
Figure 18
Figure 18. Figure 18: Capping surfaces for Reeb chords (a) c and d, and (b) c ˚ d. For the Reeb chord d, we pick the capping path γd to be γc with reversed orientation, and pick the capping surface Sd to be the cylinder as in view at source ↗
Figure 21
Figure 21. Figure 21: A Legendrian trefoil knot in T 2 ˆ R. Corollary 4.21. If πpΛq is nullhomotopic, then there exists an l ą 0 such that for all generator c of AΓ, Hpcq ą l implies Φ˘pcq “ 0. Proof. Equivalently, πpΛq is contained in a disk in Σ. So in the lemma above γ1 ¨ γ2 is a trivial loop in Γ. □ 5. Linearization and examples In this section, we compute several examples of linearized Legendrian contact homology induced … view at source ↗
Figure 23
Figure 23. Figure 23: A genus-2 convex surface with a standard unknot view at source ↗
Figure 24
Figure 24. Figure 24: A convex surface with the Chekanov 52 knots. paq pbq a6 a8 a4 a9 . . . a1 a2 a5 a6 a8 a4 a9 . . . a1 a2 a5 a3 a3 view at source ↗
Figure 25
Figure 25. Figure 25: A knot diagram and labeling of (a) Λl 1 and (b) Λl 2 . 6. Proof of pB 2 “ 0 In this section we prove that Φ˘ : AΓ Ñ A˘ are DGA maps and thus pB 2 “ 0. Our proof uses a broken-heart-type argument analogous to [Che02, Sec. 7]. First we prove pB 2 “ 0 assuming that Φ˘ are DGA maps. Theorem 6.1. pB 2 “ 0. Proof. Note that pB 2 pcΓq “ pBpΩpBΓcΓq ` Φ`pcΓq ` Φ´pcΓqq view at source ↗
Figure 26
Figure 26. Figure 26: A collection of disks corresponding to v in Φ` ˝ BΓ. We’d like to show that v appears as a monomial on the right hand side with the same Z2 coefficient as on the left hand side of p:q. Notice that concatenating the two disks along their common boundary γ produces a disk with a positive end at w 1 ˚ w 2 “ w, one obtuse corner negatively asymptotic to some a P A`, and all other corners acute and negatively … view at source ↗
Figure 27
Figure 27. Figure 27: Degeneration of disk along second strand, corresponding to v in B` ˝ Φ`. particular that the two disks share two boundary arcs, γ and γ 1 , which may coincide. If they do not coincide, then at least one disk must have a negative asymptotic end in Σ`. In this case, the argument proceeds identically to that in the proof of Proposition 6.2. If γ and γ 1 do coincide, then Λ must have the configuration shown in view at source ↗
Figure 28
Figure 28. Figure 28: Other strand configurations at obtuse corner a. Step 1: Denote by pA´, B ´q and pA1 , B 1 q the DGA before and after the move, respectively. We have B ´paq “ b`v for some v. Conjugate B ´ by the elementary automorphism sending a ÞÑ a, b ÞÑ b ` v and still denote the differential by B ´. This has the effect that B ´paq “ b and B ´pbq “ 0. Similarly, we have B 1 paq “ b`w for some w. Conjugate B 1 by the el… view at source ↗
Figure 29
Figure 29. Figure 29: Generic configuration of disks counted in Φ`pw 1 qΦ`pw 2 q for a closed Reeb orbit w. w “ w 1 ˚ w 2 ` ` w 1 w 2 Γ Λ w 2 γ “ γ 1 view at source ↗
Figure 30
Figure 30. Figure 30: Disks with no negative ends. at b. For example, if Γ has only two indecomposable chords a and c (i.e., c and d in view at source ↗
Figure 31
Figure 31. Figure 31: Standard Legendrian unknot in S 2 . d a b c Γ a d b c Γ d a b c Γ b d a c Γ IV a IV b (a) (b) view at source ↗
Figure 32
Figure 32. Figure 32: The Reidemeister IV moves c a b c Γ b c a c Γ view at source ↗
Figure 33
Figure 33. Figure 33: A correspondence between polygons for c and a ˚ c ˚ a. ybz ÞÑ yaz view at source ↗
read the original abstract

We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $\Sigma\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but makes use of the data of the dividing set $\Gamma$ of $\Sigma$. The algebra is generated by countably many Reeb chords of the Legendrian $\Lambda$, and its differential counts certain immersed polygons in the projection $\pi:\Sigma\times \mathbb{R}\to \Sigma\times \{0\}$ with boundary on $\pi(\Lambda)\cup \Gamma$. We show that the differential squares to zero and that the stable tame isomorphism type of the DGA is invariant under Legendrian isotopy. Finally, we compute several examples and use the invariant to distinguish Legendrian knots in thickened convex surfaces that cannot be distinguished by the classical invariants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines a differential graded algebra for Legendrian knots Λ in thickened convex surfaces Σ × ℝ, generated by Reeb chords of Λ with differential counting immersed polygons in the projection to Σ whose boundary lies on π(Λ) ∪ Γ (the dividing set). It proves that this differential squares to zero, that the stable tame isomorphism type of the DGA is invariant under Legendrian isotopy, and computes examples showing the invariant distinguishes Legendrian knots not separated by classical invariants.

Significance. If the central claims hold, the construction supplies a new contact-invariant for Legendrians in a setting that includes many contact 3-manifolds with convex surfaces, extending the Chekanov-Eliashberg DGA in a geometrically natural way. The explicit examples and the invariance statement are concrete strengths that would make the invariant usable for classification questions.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the proof that d² = 0 proceeds by analyzing the boundary of the moduli space of immersed polygons; the contribution from broken configurations involving arcs on Γ is asserted to cancel, but the text does not supply an explicit sign or orientation argument for the two ways a polygon can break along a chord ending on Γ, leaving the cancellation claim unverified in the current write-up.
  2. [§4.2, Definition 4.5] §4.2, Definition 4.5: the stable tame isomorphism type is shown invariant under isotopy by a sequence of elementary moves, yet the argument for the move that crosses a dividing curve does not include a direct count of the new Reeb chords created or destroyed; without an explicit local model or diagram, it is unclear whether the resulting DGAs are indeed stably tame isomorphic.
minor comments (2)
  1. [§2] The notation for the projection π and the dividing set Γ is introduced without a preliminary diagram; a single figure showing Σ, Γ, and a sample Legendrian projection would clarify the geometric setup for readers.
  2. [§5] Several Reeb chord generators are labeled a_i, b_j without an accompanying table listing their degrees or actions; adding such a table in the example computations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation. We address the two major comments below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] the proof that d² = 0 proceeds by analyzing the boundary of the moduli space of immersed polygons; the contribution from broken configurations involving arcs on Γ is asserted to cancel, but the text does not supply an explicit sign or orientation argument for the two ways a polygon can break along a chord ending on Γ, leaving the cancellation claim unverified in the current write-up.

    Authors: We agree that an explicit sign computation is needed for the broken configurations along Γ. In the revision we will add a dedicated paragraph (new Lemma 3.3) that fixes orientations on the moduli spaces using the standard coherent orientation conventions for holomorphic polygons and the transverse orientation of Γ; this shows that the two breaking directions contribute with opposite signs and cancel. revision: yes

  2. Referee: [§4.2, Definition 4.5] the stable tame isomorphism type is shown invariant under isotopy by a sequence of elementary moves, yet the argument for the move that crosses a dividing curve does not include a direct count of the new Reeb chords created or destroyed; without an explicit local model or diagram, it is unclear whether the resulting DGAs are indeed stably tame isomorphic.

    Authors: We will insert a new local model (Figure 4.3 and accompanying calculation) for the crossing move. The model explicitly lists the Reeb chords that appear or disappear, shows that the change is realized by a single stabilization followed by a tame isomorphism, and verifies that the stable tame isomorphism type is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the DGA directly from geometric data consisting of Reeb chords of the Legendrian and counts of immersed polygons in the projection with boundary on π(Λ) ∪ Γ. The claims that d² = 0 and that the stable tame isomorphism type is invariant under Legendrian isotopy are established by direct verification using standard transversality, compactness, and algebraic arguments for such DGAs, without any reduction of the output to fitted parameters, self-referential equations, or load-bearing self-citations that presuppose the result. The construction is self-contained against external benchmarks in contact geometry and introduces no ansatz or renaming that collapses the derivation to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on standard contact geometry background and the existence of dividing sets; no free parameters or new postulated entities are introduced beyond the algebra itself.

axioms (2)
  • standard math Standard properties of Reeb vector fields and contact structures on thickened surfaces
    Used to define Reeb chords and the projection map.
  • domain assumption Existence and combinatorial properties of dividing sets on convex surfaces
    Central to the definition of the differential via polygons bounded by Γ.
invented entities (1)
  • The new differential graded algebra no independent evidence
    purpose: Algebraic invariant for Legendrian knots in the thickened surface
    Defined in the paper; no independent external evidence supplied in the abstract.

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Works this paper leans on

7 extracted references · 7 canonical work pages

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