Weak Relative Calabi-Yau Structures for Legendrian Contact Homology

Jiajie Ma; Joshua M. Sabloff

REVIEW 3 major objections 5 minor 56 references

Weak Relative Calabi-Yau Structures for Legendrian Contact Homology

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For Legendrian knots, the linear duality long exact sequence for Legendrian contact homology lifts to a weak right relative Calabi-Yau structure of dimension 2 on the positive augmentation category.

desk verdict A serious, carefully written preprint that upgrades linearized LCH duality to a weak relative CY structure; the main architecture is credible, but the key homotopy Lemma 7.2 is not fully written out and the deferred cases are exactly where a referee should push. read the letter →

arxiv 2509.02485 v1 pith:XC2UFX6A submitted 2025-09-02 math.SG

Jiajie Ma , Joshua M. Sabloff This is my paper

classification math.SG MSC 53D4253D3757K1057K33

keywords LegendrianknotcontacthomologyaugmentationcategoryCalabi-YaustructureA-infinitybimoduledualityexactsequenceChekanov-EliashbergDGA

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the familiar Poincaré-Lefschetz-style duality for linearized Legendrian contact homology is not just a linear fact: it is the visible shadow of a higher algebraic structure. Specifically, for a Legendrian knot in standard contact R3 with a simply perturbed positive augmentation category, the projection functor to the circle category carries a weak right relative Calabi-Yau structure of dimension 2. This packages the duality isomorphism between the negative and positive augmentation bimodules together with the circle bimodule into a homotopy-commutative diagram of exact triangles. A sympathetic reader should care because the result places the classical duality exact sequence inside the same Calabi-Yau framework used for wrapped Fukaya categories and mirror symmetry, and because it offers a template for nonlinear duality in contact homology.

What carries the argument

The load-bearing mechanism is a conical pair (Aug+(Λ) π→ C(Λ), N[−1] ρ→ M−), in which the positive augmentation bimodule M+ is realized as the mapping cone of ρ. The proof constructs a very weak relative Calabi-Yau structure on ρ—quasi-isomorphisms η, θ, η′ fitting into a homotopy-commutative diagram—using counts of thin and thick holomorphic disks in (s,r)-copies of the 2-copy and separated 2-copy of Λ, identified by enriched-disk projection and lifting lemmas. A purely algebraic upgrade (Proposition 4.4) then converts the very weak structure into the weak right relative Calabi-Yau structure on π_C, with θ supplied by a Morse-theoretic Poincaré duality for the circle.

What would settle it

Check the homotopy equation for inputs (x+, a∨, x+) on the figure-eight knot from Example 6.5: the terms in (7.18) must cancel by the enrichment rules of Lemma 6.27. An explicit enumeration of the doubly enriched disks with mixed enrichment at t−1 that fail to cancel would refute Theorem 1.1; this is a finite disk count.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1.1: the projection functor π_C: Aug+(Λ) → C(Λ) admits a weak right relative Calabi-Yau structure of dimension 2. Concretely, there is a homotopy-commutative isomorphism of exact triangles of Aug+(Λ)-bimodules whose vertices are M+ (the diagonal bimodule of Aug+(Λ)), N (the pullback of the diagonal bimodule of the circle category), and M− (the negative augmentation bimodule). The leftmost vertical map is a quasi-isomorphism η: M∨−[−2] → M+ built from the separated 2-copy, and the theorem states that this higher morphism, together with the circle-category duality θ: N∨[−1] → N, organizes the classical duality long exact sequence and its m

Load-bearing premise

The whole structure hinges on the disk-counting enrichment constraints in the (s,r)-copy of the separated 2-copy—especially the allowed multiplicities around the base point t versus t−1—so a single miscount would make the telescoping sums that prove homotopy commutativity fail.

Editorial extensions

If this is right

The classical duality long exact sequence of Theorem 2.8 is recovered by reducing Diagram (1.1) to linear morphism spaces, so the new structure strictly generalizes the old duality.
The quasi-isomorphism η: M∨−[−2] → M+ gives a way to compute linearized contact homology of the positive category from the negative one in the derived category of bimodules.
The same conical-pair method should apply to any horizontally displaceable Legendrian in a 1-jet space, subject to higher-dimensional perturbation and mixed-puncture checks.
Evidence in the paper suggests the weak structure should lift to a strong relative Calabi-Yau structure, connecting with relative Ginzburg algebra and microlocal sheaf constructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the enrichment constraints survive a coefficient lift, the whole theorem is expected to hold over Q and any commutative ring, not just F2, since the obstruction is combinatorial rather than characteristic-dependent.
A natural testable consequence is that the homotopy category of Aug+(Λ) becomes a relative 2-Calabi-Yau category over the circle category, which would place Legendrian knot invariants inside the same relative CY framework used for microlocal sheaves and wrapped Fukaya categories.
The explicit disk counts used to prove Lemma 7.2 could be turned into a computational algorithm on small knots, producing the first concrete examples of relative CY structures in Legendrian contact homology.
Generalizing the separated 2-copy construction to multiple mixed punctures would likely produce higher-dimensional relative CY structures and, ultimately, a categorified duality for Legendrian submanifolds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The takeaway: this is a real result, not a claim outrunning its evidence. Theorem 1.1 is new—no prior statement gives a weak right relative CY structure of dimension 2 on the projection Aug+(Λ) → C(Λ). The paper builds cleanly on the augmentation-category framework from [43] and the Brav–Dyckerhoff relative CY language, and the overall proof architecture is coherent: an algebraic bootstrap (Proposition 4.4) plus geometric bimodule constructions, with the hard disk-counting done in terms of enriched disks rather than left as a black box.

What the paper does well: it is unusually honest about scope. F2 coefficients, the simply perturbed perturbation model, and the conjectural perturbation independence (Remark 6.10) are all stated plainly. It also acknowledges Chen’s overlapping independent work, which is the right thing to do and is not a mark against novelty. The explicit formulas for θρ∨ and πNη, and the detailed telescoping arguments in Claims 7.4–7.9, are genuine mathematical work.

The real soft spot is Lemma 7.2. The homotopy H is verified only for selected inputs; the remaining cases are deferred with “symmetric or slightly generalized arguments with no new ideas necessary” (§7.2.2, §7.2.3). The stress-test concern is not hypothetical: if any omitted case has a non-canceling term—say an x+ input with t−1 mixed enrichment violating the constraints in Lemma 6.27—then the homotopy fails, Lemma 4.3 cannot be applied, and Theorem 1.1 does not follow. I do not think that failure is likely; the written cases show the pattern, and the enrichment constraints are explicit. But “likely” is not the same as “checked.” The thin-disk classifications in Lemmas 5.5 and 5.12 are also asserted via “direct combinatorial enumeration” with representative figures; those feed directly into the telescoping sums, so a miscount there would silently break the proof.

Two smaller points. Proposition 6.25 proves quasi-isomorphism via an isotopy and acyclicity argument rather than an explicit inverse; that is a legitimate strategy, but it makes one of the key maps less transparent. And the use of [15,47] as an ingredient is fine—those are prior theorems, not the conclusion.

This deserves a serious referee. The claim is important and the architecture is credible; the referee should specifically check the omitted cases in Lemma 7.2 and the enumerations in Section 5. I would send it to an expert, not desk-reject.

Referee Report

3 major / 5 minor

Summary. The paper proves (Theorem 1.1) that for a Legendrian knot Λ in the standard contact R^3 and a simply perturbed positive augmentation category Aug+(Λ) over F2, the projection functor π_C: Aug+(Λ) → C(Λ) to the circle category admits a weak right relative Calabi-Yau structure of dimension 2. This means, at the level of A∞ bimodules, a homotopy-commuting isomorphism of exact triangles (Diagram 1.1) whose linear shadow recovers the classical Ekholm–Etnyre–Sabloff duality long exact sequence. The proof has three strands: (i) an algebraic bootstrap (Section 4) showing that a very weak relative CY structure on ρ : N[−1] → M− over a conical pair yields a weak relative CY structure on π (Proposition 4.4); (ii) a geometric analysis of Reeb chords and immersed disks in n-copies, (s,r)-copies, and separated 2-copies (Section 5), feeding the definitions of Aug±(Λ), C(Λ), and the bimodules M+, M−, N (Section 6); and (iii) a case-by-case verification that θρ∨ and π_Nη are homotopic (Lemma 7.2, Section 7), which supplies the very weak CY structure (Proposition 7.10). The paper works over F2, assumes a specific 'simply perturbed' Morse perturbation, and explicitly acknowledges concurrent independent work by Chen [7].

Significance. If the result holds, it is a genuine structural advance: it promotes the linear duality long exact sequence for Legendrian knots to a nonlinear A∞-bimodule statement, in parallel with Chen's independent LSFT approach and with microlocal-sheaf relative CY structures of Kuo–Li. The algebraic reduction in Proposition 4.4 (very weak → weak relative CY via a conical pair, with explicit chain models) is clean, reusable, and a solid contribution in its own right. The paper is unusually explicit about its geometric inputs: disk identifications, enrichment bookkeeping (m0, n0 constraints), and a fully worked figure-eight-knot example (Example 6.26) verifying the quasi-isomorphism η. It is also honest about its limitations, flagging the conjectural independence of the perturbation scheme (Remark 6.10) and the reliance on the classical duality [15,47]; the latter is a legitimate ingredient, not a circularity. The principal risk is the completeness of the combinatorial verification, which is substantial and load-bearing.

major comments (3)

[§7.2, Lemma 7.2] The proof of the key homotopy θρ∨ ∼ π_Nη is incomplete. Claims 7.4–7.9 verify the homotopy equation only for inputs a∨, (a∨,a+j), (a∨,y+), (a∨,x+), (x+,a∨,a+j), and (x+,a∨,x+). All remaining inputs are deferred: §7.2.2 states that the rest follow from 'symmetric or slightly generalized arguments with no new ideas necessary,' and §7.2.3 does the same. The cancellations are delicate: they depend on the enrichment constraints m0 ≤ 1, n0 = 0 for t and m0 = 0 for t−1 in Lemma 6.27/eq. (6.9), and on the telescoping sums (7.12)–(7.15) matching the terms in (6.9), (7.4), (7.5). A non-canceling term in an omitted case—e.g. an input beginning with x+ with mixed t−1 enrichment—would invalidate H, so Lemma 4.3 would not apply and Proposition 7.10 (hence Theorem 1.1) would not follow. The omitted cases must be written out or reduced by an explicit symmetry with the enrichment bookkeeping tracked.
[§5.3–5.4, Lemmas 5.5, 5.12] The thin-disk classifications are load-bearing but only partially verified. Lemma 5.5's proof is 'direct combinatorial enumeration, half of which is carried out in Figure 11'; Lemma 5.12's proof is a 'straightforward combinatorial enumeration' in Figure 16; Lemma 5.10 inherits from Lemma 5.5. These classifications underlie the characterizations of M+, N, ρ∨, and π_Nη (Lemmas 6.11, 6.27) and enter the telescoping sums of §7.2. The unspecified 'half' means the reader cannot identify which configurations remain unchecked. Since these statements feed the enrichment constraints in (6.9), the paper should provide the complete enumeration or a formal reduction to the n-copy classification (Lemma 5.1).
[§7.2.1, p. 60] The general Case-1 inputs are not proved. After Claims 7.4–7.5, the text states that 'a similar argument using disks with more pure enrichments works for longer inputs' without specifying the resulting formula. Together with the deferrals in §7.2.2–7.2.3, the infinite family of A∞ homotopy equations δ(H) = π_Nη + θρ∨ is not established for arbitrary r,s. Because H is used to construct the very weak CY structure via Lemma 4.3, the full verification is a necessary part of the proof, not a presentation choice.

minor comments (5)

[Remark 6.10] The conjectural dependence of all bimodule structures on the simply perturbed perturbation scheme should be flagged in the Introduction. As stated, Theorem 1.1 is a statement about a particular perturbation; the reader should be told that independence of this choice is open.
[Proposition 6.25] The three-sentence proof would benefit from an explicit statement of why the horizontal isotopy acts by a quasi-isomorphism on the full A∞ bimodule cM, and how chain-level acyclicity of the linearized separated 2-copy implies the quasi-isomorphism of η (via Proposition 3.13).
[Remark 5.13] The thick disks with z ∈ bR_q and w ∈ bR_p are not identified; since they project to disks with two positive corners, the paper should state explicitly that they do not contribute to the maps used in Lemma 6.27 and §7.2, or account for them in the homotopy verification.
[§5.1.2, Definition 5.2] The thick/thin dichotomy is asserted by a 'geometric argument.' For the separated 2-copy, with its p/q chord types and disks with two positive corners, the dichotomy should be stated precisely so that 'thin' and 'thick' are exhaustive in each of the three cases of Lemma 5.14.
[Throughout] Minor issues: 'the the 2-copy' (§5.3.1); the phrase 'simple perturbed' appears alongside 'simply perturbed'; the degrees of a∨, x∨, y∨ are not recorded in Lemmas 6.18 and 6.27; Example 6.26 checks the quasi-isomorphism η only on Hom(ε1, ε1) and should state the analogous check for the other object pairs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the weak relative CY structure is derived from geometric disk counts and standard invariance inputs, not from its own conclusion.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The central claim (Theorem 1.1) is obtained by applying the algebraic Proposition 4.4 to a very weak relative CY structure constructed in Section 7. The key step, Lemma 7.2, verifies the homotopy θρ∨ ∼ πNη by explicit telescoping-sum cancellations over doubly enriched disks. The cancellations use only the augmentation equations ε∘∂=0 and the disk-counting characterizations in Lemmas 6.18, 6.27, and Proposition 7.1; they do not presuppose the homotopy or the CY structure. The quasi-isomorphism η : M∨−[−2] → M+ is proved in Proposition 6.25 by showing the separated 2-copy bimodule is acyclic after a Legendrian isotopy and invoking invariance of (bi-)linearized Legendrian contact homology. This uses the standard invariance of linearized LCH, not the duality long exact sequence (Theorem 2.8) that the paper aims to generalize; that sequence is recovered only afterward in Remark 7.11. The morphism θ is defined explicitly in Proposition 7.1 from the Morse theory of the circle category, and ρ∨ is defined geometrically in Lemma 6.18. The citations to [15,47], which include the second author, are used for context and for the separated 2-copy technique, but the proof does not rely on the linear duality result as a black-box input; it proves the needed quasi-isomorphism independently. The augmentation-category framework is imported from [43], which has no author overlap. The paper's own flagged limitations — Remark 6.10 on conjectural perturbation independence, the deferred 'symmetric' cases in §7.2.2 and §7.2.3, and the 'direct combinatorial enumeration' assertions in Lemmas 5.5 and 5.12 — are genuine completeness or exposition risks, but they are not circularity: no equation is defined in terms of the conclusion, and no fitted parameter is later renamed as a prediction. The derivation does not reduce, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No free parameters are fitted: the claims depend on structural choices (F2 coefficients, finite-dimensional morphism spaces, a specific simply perturbed perturbation scheme) rather than on numbers chosen to make a fit work. The axioms are standard results from the field, imported from [6, 2, 43, 15, 47], plus the paper's own geometric disk-identification lemmas, which are proven by enumeration but not formalized. The invented algebraic objects (C(Λ), N, M−, ρ) are constructed from defined geometry, not postulated, and each has an independent check at the linear level.

assumptions (6)

standard math Chekanov-Eliashberg DGA: well-defined differential, squares to zero, Legendrian isotopy invariance over F2 (Theorem 2.1, [6, 20, 40])
Relied on throughout as the foundation of LCH; accepted from citation.
domain assumption Augmentation categories Aug±(Λ) are A∞ categories, invariant under perturbation and Legendrian isotopy (Theorem 6.2, [2, 43])
The entire bimodule theory is built on this; perturbation independence of the categories is cited, not reproven.
domain assumption Finite-dimensionality of all morphism spaces (stated in Section 3)
Needed for Proposition 3.13, that quasi-isomorphisms are A∞ homotopy equivalences, the bootstrap used repeatedly.
domain assumption Classical linear duality exact sequence (Theorem 2.8, [15, 47]) used to justify acyclicity of the separated 2-copy bimodule in Proposition 6.25
The duality quasi-isomorphism η: M∨−[−2] → M+ is the linchpin; its proof invokes invariance of (bi)linearized LCH under Legendrian isotopy.
ad hoc to paper Disk identification lemmas in Section 5 (Lemmas 5.1, 5.3, 5.5, 5.8, 5.10-5.14): classifications of thin and thick disks in n-copies, (s,r)-copies, and separated 2-copies
Proven by combinatorial enumeration and figures, not formalized; the load-bearing geometric input for the bimodule structure maps and the homotopy cancellation arguments.
ad hoc to paper Simply perturbed configuration: Morse function f with one max and one min adjacent in the Lagrangian diagram, unique basepoint just before the max, and specific placement of critical points in (s,r)-copies (Figure 10)
The statement of Theorem 1.1 depends on this genericity and perturbation model; the bimodule structure maps are only defined in this model, and Remark 6.10 notes perturbation independence is conjectural.

invented entities (2)

Circle category C(Λ) and circle bimodule N independent evidence
purpose: Target of the projection functor π_C; N ≅ π∗C(Λ)∆ serves as the boundary term in the relative CY structure
Constructed as a quotient of Aug+ by the Reeb-chord subcategory; at the linear level it recovers Morse cohomology of S1, and the bimodule quasi-isomorphism θ: N∨[−1] → N in Proposition 7.1 is checked by direct computation.
Negative augmentation bimodule M− and the morphism ρ: N[−1] → M− independent evidence
purpose: Provide the mapping-cone model M+ ≅ A∆ ≅ Cone(ρ), the conical pair that feeds the algebraic bootstrap Proposition 4.4
M− is identified with the length-filtration submodule M_+^{Reeb} (Proposition 6.14) and with the Reeb subcategory Aug_+^{Reeb} ≅ Aug−; at the linear level it yields the classical LCH(−) duality groups.

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Pith. "Pith review of Weak Relative Calabi-Yau Structures for Legendrian Contact Homology." pith.science (2026). https://pith.science/paper/XC2UFX6A

@misc{pith2026250902485,
  author       = {Pith},
  title        = {Pith review of: Weak Relative Calabi-Yau Structures for Legendrian Contact Homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XC2UFX6A}},
  note         = {Machine review of arXiv:2509.02485}
}

abstract

Legendrian Contact Homology (LCH) and its augmentations are important invariants of Legendrian submanifolds, and for Legendrian knots in the standard contact 3-space in particular. We increase understanding of the algebraic structure of LCH by generalizing the duality isomorphism and long exact sequence for linearized LCH for Legendrian knots to a weak relative Calabi-Yau structure for $A_\infty$ bimodules over the positive augmentation category.

Figures

Figures reproduced from arXiv: 2509.02485 by the authors.

**Figure 1.** (a) Reeb signs at a crossing of the Lagrangian diagram of a Legendrian knot Λ, with dots denoting positive corners and a lack of decoration denoting a negative corner. (b) A schematic picture of a disk in ∆Λ(a, b1b2t). The differential ∂ on A is defined by counting immersed disks in the Lagrangian diagram πxy(Λ), which stand in for pseudo-holomorphic disks in the symplectization R × R 3 ; see [14, 20] for more on t… view at source ↗

**Figure 2.** The Lagrangian diagram of a Legendrian figure eight knot. a2, a3 have grading −1. The differential is given by: ∂a1 = ∂a2 = ∂a3 = ∂a5 = ∂t±1 = 0 ∂a4 = a2 + a3 + a2a1a3 ∂a6 = t + a5 + a1a3a5 ∂a7 = 1 + a5 + a5a2a1. The Chekanov-Eliashberg DGA is filtered by the length of the Reeb chords. This follows from an application of Stokes’ Theorem. Stated precisely, we have: Lemma 2.3 ([6, Lemma 6.1]). Let ℓ(x) represent the … view at source ↗

**Figure 3.** A term b in the linearized differential ∂ ε 1 (a) arises from a disk in ∆Λ(a, a1, . . . , an, b, b1, . . . bm) with ε(ai) = 1 = ε(bi). To turn T A¯ ∗ into a DG co-algebra (DGCA), we take the adjoint δε of ∂ ε with respect to the pairing ⟨,⟩: ⟨δεx ∗ , y⟩ = ⟨x ∗ , ∂ε y⟩. See [28, §1] for a concise introduction to DG co-algebras (DGCAs). Using Proposition 2.4, we obtain the following. Proposition 2.6. The map δε : T A¯… view at source ↗

Figures from the paper (20 more)

**Figure 4.** Figure 4: When all of the corners in the dotted parts of the boundary are mapped to 1 by ε, this disk contributes a ∨ to mk ε (a ∨ 4 , . . . , a∨ 1 ). order), and possibly other negative corners that the augmentation ε maps to 1; see [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: (a) For every crossing of Λ, there are n 2 crossings of Λn . (b) For every critical point of the perturbing function f, there are 1 2 n(n − 1) crossings of Λn . 5.1. The n-Copy of a Link. We begin by setting notation for the Reeb chords of the n-copy of a Legendrian li…

**Figure 6.** Figure 6: The thin disks in the n-copy of a Legendrian link take one of these forms, as listed in Lemma 5.1. We will use the red dots to indicate crossing information in this and subsequent figures that display disks in Lagrangian diagrams. In particular, a disk in ∆Λn (a, a) is…

**Figure 7.** Figure 7: The disk at left is an enriched disk in ∆3 Λ (a, a1t −1 ) with w = {1, 2, 2}. The disk at right is its lift to a disk in ∆Λ4 (a 14, a12 1 x 23x 34(t 4 ) −1 ), following Lemma 5.3. how s ◦ u induces an enriched disk (ˇu, w(u)) in ∆n−1 Λ (ˇa, aˇ), with enrichment w(u) to…

**Figure 8.** Figure 8: A Lagrangian diagram of the (2, 3)-copy Λ2,3 of a 2-component link Λ. Label the components of the s-copy from bottom to top and those of the r-copy from top to bottom. Finally, place base points on Λs,r just before the maxima of the functions fi , as in the case of an …

**Figure 9.** Figure 9: A thick disk in (s, r)-copy form in the set ∆Λ(z 12|23, x11|21(t 1|1 ) −1w 12|11t 2|1x 22|12a 22|23). the (s, r)-copy. When working with the (s, r)-copy, we say a pair (z, awb) is in (s,r)-copy form if • z ∈ R12|sr , • w ∈ R12|11 , • aik ∈ R11|k+1,k and ak are words in…

**Figure 10.** Figure 10: The construction of a simply perturbed (s, r)- copy of the 2-copy near the critical points. Legendrian knot, allowing us to identify disks in the (s, r)-copy with disks in the knot, not just the 2-copy link. 5.3.1. The 2-Copy Construction. For each crossing a of Λ, th…

**Figure 11.** Figure 11: The thin disks in the 2-copy listed in the second column of Lemma 5.5. The disks listed in the first column are similar, but pass around larger portions of the knot as in [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 12.** Figure 12: The disk at left is an enriched disk in ∆2 Λ (z, aw) with pure enrichments w1 = {1} and w2 = ∅. The disk at right is its lift to a disk in ∆Λ 2,2 (z 12|21, a11|21w 12|11), following Lemma 5.8. z a z 11 z 12 z 22 z 21 a 21 a 22 a 12 a 11 x 12 y 12 [PITH_FULL_IMAGE:fig…

**Figure 13.** Figure 13: The disk at left is an enriched disk in ∆2 Λ (z, at−1 ) with pure enrichments w1 = {1} and w2 = {0}. The disk at right is its lift to a disk in ∆Λ 2,2 (z 12|22, a11|21x 12|11x 22|12(t 2|2 ) −1 ). on a correspondence with chords of Λ; our formulation matches that in [4…

**Figure 14.** Figure 14: The separated 2-copy of a Legendrian knot. of chords p chords and the latter type q chords, denoted by Rbp and Rbq, respectively. Note that the chords arising from the critical points of the perturbation are q chords. We also use these letters to distinguish these two…

**Figure 15.** Figure 15: In Lemma 5.10, the configuration of the crossing signs for the x, y, and q generators is the same as those for the x, y, and a ij (i ≥ j) generators in [PITH_FULL_IMAGE:figures/full_fig_p038_15.png]

**Figure 16.** Figure 16: The thin disks in the 2-copy listed in Lemma 5.12. (x 12|11, t−1 a 11|11p 12|11t) (x 12|11, t−1 p 12|11a 22|11t)(5.5a) (x 12|21, t−1 a 11|21p 12|11t) (x 12|12, t−1 p 12|11a 22|12t)(5.5b) (x 12|12, t−1 a 11|11p 12|11tx22|12) (x 12|12, t−1 p 12|11a 22|11tx22|12)(5.5c) (…

**Figure 17.** Figure 17: The thick disks captured by Lemma 5.14(1) on the top and (2) on the bottom come from the same underlying enriched disk in ∆1,0 Λ (a1 , a2a3 ) with pure enrichment w1 = {1} and w2 = ∅. At top, the enriched disk corresponds to a disk in ∆Λb2,2 (q 12|21 1 , a 12|21 2 q 1…

**Figure 18.** Figure 18: The thick disks captured by Lemma 5.14(3). The enriched disk at left lies in ∆1,0 Λ (a1, a2t −1 ) with pure enrichments w1 = {1} and w2 = {0} corresponds to the disk at right in ∆Λb2,2 (y 12|22, x11|21(t 1|1 ) −1p 12|11 1 a 22|12 2 ). z ∈ Rbp, w ∈ Rbp: The stick-toget…

**Figure 19.** Figure 19: Enriched disks that contribute a + to m4 +(a + 4 , . . . , a+ 1 ) (left) and to m5 +(a + 5 , x+, x+, a+ 2 , a+ 1 ) (right). Enriched disks on the left, but not the right, side contribute to m4 −(a − 4 , . . . , a− 1 ). Enriched disks yield additional terms mk +(a + k …

**Figure 20.** Figure 20: Doubly enriched disks that contribute z + to n r|s M+ ( ⃗b, w+, ⃗a) (left) and to n r|s M+ ( ⃗b, t±1 , ⃗a), with n0 and m0 x + generators before and after w + (right); recall that m0 = 0 = n0 if w = t. Doubly enriched disks on the left side contribute z − to n r|s M−…

**Figure 21.** Figure 21: A comparison of the perturbation functions (f, f, f) at left and (−f, f, −f) at right for the separated 2-copy. By isotoping the boxed y chords for the s- and rcopies in the (−f, f, −f) perturbation leftwards around the knot, we recover the (f, f, f) perturbation. te…

**Figure 22.** Figure 22: The disk in Λ pictured has multiple possible mixed enrichments, each of which could contribute to n 0|0 M− (a ∨), depending on the value the augmentations ε 1 1 and ε 2 1 take on the non-enriched negative corners. The corners are labeled with these augmentation value…

**Figure 23.** Figure 23: The disk in Λ pictured has multiple possible enrichments. The disks in the top row could contribute to n 0|1 M∨ − (a ∨, a+ j ), depending on the value the augmentations take on the non-enriched negative corners; here, j = 3. The disk at bottom left could contribute to…

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