Non-decomposable Lagrangian cobordisms between Legendrian knots

Daniel Kom\'arek; Roman Golovko

REVIEW 2 major objections 5 minor 16 references

Non-decomposable Lagrangian cobordisms between Legendrian knots

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For every genus, non-decomposable Lagrangian cobordisms exist

desk verdict First non-decomposable Lagrangian cobordisms of arbitrary positive genus, built from a clean Livingston-style obstruction; the only real caveat is that the central Lagrangian-concordance step is outsourced to unpublished preprints. read the letter →

arxiv 2511.08731 v3 pith:Z7EJS726 submitted 2025-11-11 math.SG

Roman Golovko , Daniel Komárek This is my paper · ORCID

classification math.SG MSC 53D1253D42

keywords LegendrianknotsLagrangiancobordismsdecomposableribbontwo-foldbranchedcoverscriticalpointestimatesdeterminantstabilizations

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 for any chosen genus g > 0 there are Legendrian knots in the standard contact three-sphere connected by a Lagrangian cobordism of genus g that cannot be built from the two elementary pieces — a birth (0-handle) and a surgery (1-handle). The input is a Legendrian knot whose determinant is divisible by an odd prime and which admits a decomposable concordance from the standard tb = −1 unknot. By taking connected sums and attaching a standard genus-g piece, the authors obtain a cobordism whose ends are stabilized versions of the original knot and the unknot. A critical-point estimate on two-fold branched covers forces the cobordism to contain at least one index-2 critical point, so it is not ribbon and hence not decomposable. This gives non-decomposable Lagrangian cobordisms of arbitrary positive genus.

What carries the argument

The carrying mechanism is the pair consisting of (i) the two-fold branched cover homology group H_1(Σ_2(K); F_p), whose dimension is controlled by the determinant of K, and (ii) the critical-point inequality c_2(Σ) ≥ (β_1(Σ_2(K_0); F_p) − β_1(Σ_2(K_1); F_p))/2 − g(Σ) for any connected Morse cobordism Σ. In this paper the inequality is fed with K_0 = Λ^n (whose branched-cover homology is ≥ n because p divides det(Λ)) and K_1 = U^n (whose homology is 0). The positive difference amplifies with n, so taking n > 2g forces at least one index-2 critical point in the constructed cobordism, which blocks decomposability.

What would settle it

For a concrete case, e.g., a Legendrian pretzel knot with det = 9, attempt to present the constructed genus-1 cobordism as a sequence of elementary moves; if such a presentation exists, the non-decomposability claim is false. A more direct check is to compute dim_{F_3} H_1 of the two-fold branched cover of the stabilized connected sum and verify the lower bound n, since the whole obstruction rests on that count.

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Extended reading notes

Core claim

The central claim is Theorem 1: let Λ be a Legendrian knot in the standard contact S^3 such that an odd prime p divides det(Λ) and there is a decomposable Lagrangian concordance from the tb = −1 unknot U to Λ. Then for every g > 0 there are stabilized connected sums Λ^n and U^n and a genus-g Lagrangian cobordism from Λ^n to U^n that is non-decomposable. The proof reverses the concordance, uses an h-principle and an approximation theorem to make it Lagrangian, and concatenates a genus-one cobordism from a positive-then-negative stabilization. Non-decomposability comes from a critical-point estimate on two-fold branched covers: p divides det(Λ), so the branched-cover homology of Λ^n is at leas

Load-bearing premise

The construction depends on the cited step in §3 that a reversed decomposable concordance can be isotoped to a totally real concordance and then approximated by a genuine Lagrangian concordance; if that approximation is unavailable, the cobordism L_g does not exist.

Editorial extensions

If this is right

For any Legendrian knot with det divisible by an odd prime and a decomposable concordance from the unknot, the construction produces non-decomposable Lagrangian cobordisms in every genus.
The examples include the Legendrian pretzel knots P(3,−3,k) with determinant 9, their connected sums with any concordance from the unknot, and any ribbon concordance realized as a decomposable Lagrangian concordance.
The constructed cobordisms are non-ribbon, so they contain an actual index-2 critical point; they are not merely non-decomposable in a subtle sense.
The condition n > 2g is the only restriction on the genus; increasing the number of connected sum copies allows arbitrarily large genus.

Reading between the lines

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

The connected-sum amplification gives a general recipe: any concordance from a knot whose determinant has an odd prime factor can be converted into non-decomposable cobordisms of all genera, so the phenomenon is not confined to the specific examples listed.
If a version of the critical-point inequality existed for the prime 2, the construction could extend to knots whose determinant is even; the paper only uses odd primes.
The authors note that removing the stabilization of the ends is difficult because a Whitehead-doubling trick used elsewhere does not commute with connected sums; a different way to control branched-cover homology could remove that restriction.
The h-principle/approximation step in §3 is cited rather than demonstrated in this paper; if that step fails for a given input, the existence of the cobordism L_g itself would be in doubt even though the obstruction argument is sound.

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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 main result is genuinely new: for any genus g > 0, the authors produce non-decomposable Lagrangian cobordisms between stabilized Legendrian knots, provided the source has an odd prime divisor of the determinant and admits a decomposable concordance from the tb = −1 unknot. That answers a natural question left open after the genus-zero examples of Lin and of Dimitroglou Rizell–Golovko. The proof is not a grab-bag: the Livingston estimate is used in a way that actually works, with n > 2g forcing c2(L_g) > 0. The paper is short, well-organized, and honest about its limits.

The main ingredient is the cobordism L_g, obtained by concatenating the h-principle/approximation concordance S(T^n) with the Sabloff–Vela-Vick–Wong genus-one cobordisms. The obstruction part—determinant multiplicativity, Proposition 6, inequality (3.1), and the final inequality—is coherent and standard. I agree with the reader that the logical chain is sound.

Where I would push back is on the stress-test note: it correctly identifies the load-bearing external step. In §3 the authors say “we argue the same way as in [9, Section 2] and [8, Section 5]”, invoke the h-principle for totally real embeddings, and then cite Dimitroglou Rizell’s approximation theorem [7]. No statement of the exact hypotheses is given, and no verification that the totally real concordance T^n satisfies them. That is a real gap in exposition, but it is a gap of reliance on the authors’ own prior work, which is normal in a short note. A referee can check those preprints. I do not think it makes the paper unserious or circular: the obstruction is independent, and the paper’s own Remark 7 candidly explains why the method does not extend to fillable ends.

Minor soft spots: the number of stabilizations is left as “sufficiently many” without an explicit bound, and Proposition 6/Theorem 5 depend on the preprint [3] of Borodzik–Tröl. These are worth flagging but not fatal.

Bottom line: this is a solid contribution for symplectic/contact topologists working on Lagrangian cobordisms. It deserves a serious referee, and the referee should focus on verifying the §3 step and the precise form of the Livingston inequality. I would bring it to a reading group and would probably cite it.

Recommendation: send to peer review, with a request to check the external dependencies.

Referee Report

2 major / 5 minor

Summary. The paper claims: if Λ is a Legendrian knot in (S^3, ξ_std) such that an odd prime p divides det(Λ) and there is a decomposable Lagrangian concordance C from the tb=-1 unknot U to Λ, then for every g>0 there exist Legendrian knots Λ− and Λ+ (sufficiently stabilized connected sums of Λ^n and U^n, respectively) and a genus-g Lagrangian cobordism L_g from Λ− to Λ+ that is non-decomposable. The proof reverses the n-fold connected sum of C, uses the h-principle for totally real embeddings together with Dimitroglou Rizell's approximation theorem to obtain a stabilized Lagrangian concordance S(T^n), attaches genus-one cobordisms of Sabloff–Vela-Vick–Wong, and applies Livingston's estimate to show that L_g has a positive number of index-2 critical points. Examples are drawn from pretzel knots, connected sums, and Etnyre–Leverson realizations of ribbon concordances.

Significance. If the existence step in §3 is accepted, this is the first construction of non-decomposable Lagrangian cobordisms of arbitrary positive genus, going beyond previously known non-decomposable caps and concordances. The obstruction mechanism is attractive: it uses Livingston's branched-cover critical-point bound rather than the more elaborate invariants used in the authors' earlier work. The determinant arithmetic and the application of Proposition 6 are clean, and the paper provides several concrete families of Legendrian knots satisfying the hypotheses. The main limitation is that the crucial production of the Lagrangian concordance S(T^n) is delegated to the authors' own preprints and to a strong approximation theorem, without a statement or verification of its hypotheses in this manuscript.

major comments (2)

[§3, paragraph beginning 'Then we argue the same way as in [9, Section 2] and [8, Section 5]'] The existence of the stabilized Lagrangian concordance S(T^n) is the load-bearing premise of the proof: without it L_g is never produced, and the Livingston estimate is never applied. The paper does not state the approximation theorem from [7] nor verify its hypotheses (cylindrical ends, total reality, formal Lagrangian condition, stabilization threshold) for the T^n obtained from (C^n)^{-1}. The arguments of [8,§5] and [9,§2] are cited only as 'the same way', and these are the authors' own preprints. Please include a precise lemma with the full theorem statement and a verification for this specific T^n, or an actual proof.
[§3.3 / Theorem 5] The application of Livingston's inequality (3.1) to L_g also needs a short justification that L_g, after truncation of the cylindrical ends, is a connected Morse cobordism in [0,1]×S^3 satisfying the hypotheses of Theorem 5. This is likely standard for exact Lagrangian cobordisms, but it should be stated explicitly; if a perturbation is required, argue that c2 and g remain unchanged.

minor comments (5)

[§2, Examples C] The concordance from [11] may start at a stabilized Legendrian unknot, while condition (2) of Theorem 1 is stated for the tb=-1 unknot U. The proof only uses det=1, so the theorem can be generalized to any Legendrian unknot; please adjust the statement or add a sentence explaining the reduction.
[Theorem 1] The notation 'Λ^n(g)' is confusing; it should be Λ^{n(g)} or simply Λ^n with n chosen later as a function of g.
[§3.2] Minor wording: 'Formulas 3.1, 3.2 and 3.3' should refer to equations (3.1)–(3.3).
[§3] The notation 'S(T^n)' for the Lagrangian concordance obtained from the totally real T^n is introduced without explanation; since 'S(·)' is also used for Legendrian stabilization, a sentence clarifying the notation would help.
[References] Several key results are cited to preprints [3], [8], [9], [11]. In particular Proposition 6 is quoted from [3, Theorem 2.5]; a short proof would make the paper more self-contained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the genus-g non-decomposability is established by Livingston's independent estimate; the only flagged item is a same-author citation for the existence of S(T^n), which is a completeness risk rather than a circular reduction.

full rationale

The claimed derivation is not circular. The only step that deserves scrutiny is in Section 3, where the authors write: 'Then we argue the same way as in [9, Section 2] and [8, Section 5]... Finally, we modify T^n using the approximation result of Dimitroglou Rizell [7] and get the Lagrangian concordance S(T^n) from S(Λ^n) to S(U^n).' This is a self-citation to the authors' own preprints [8] and [9], and the existence of S(T^n) is load-bearing for the construction of L_g; moreover the hypotheses of the h-principle/approximation step are not checked in the paper. That is a legitimate completeness/correctness concern, but it is not a circularity: [8] and [9] concern Lagrangian concordances (genus zero) between stabilized Legendrian knots, not the target genus-g non-decomposable cobordisms. The non-decomposability is proved by Livingston's inequality (Theorem 5, Eq. (3.1)) applied to L_g, together with standard determinant multiplicativity and Borodzik-Tröl's Proposition 6. No fitted parameter is renamed as a prediction; no equation is identical to an input by construction; no ansatz is smuggled in by citation. The computations det(Λ^n)=det(Λ)^n, det(U^n)=1, and c2(L_g) ≥ n/2−g>0 are standard and externally supported. Hence the paper has no significant circularity; at most a minor self-citation/completeness flag applies.

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

No empirical fit parameters appear; the integer n is explicitly quantified in Theorem 1 and chosen with n>2g, so it is not a hidden degree of freedom. The central claim rests on external theorems: Livingston's estimate (Theorem 5), Borodzik–Tröl's Proposition 6, the h-principle/approximation step from [7]-[9], the SVW genus-one cobordism, and standard determinant multiplicativity. None of these is proved in the paper.

assumptions (6)

domain assumption Livingston's estimate (Theorem 5): c2(Σ) ≥ (β1(Σ2(K0);Fp) − β1(Σ2(K1);Fp))/2 − g(Σ).
Assumed from [14, Cor 5.4] and [3, Thm 2.5]; not proved in the paper. It is the core obstruction used in §3.3.
domain assumption Borodzik–Tröl Proposition 6: if p|det(K) then dim H1(Σ2(K^{#n});Fp) ≥ n; if p∤det(K) then H1(Σ2(K);Fp)=0.
Assumed from the preprint [3]; it converts the determinant condition into the Betti-number lower bound (3.2).
standard math Determinant multiplicativity under connected sum: det(K1#K2)=det(K1)det(K2), and det(K)≠0.
Used in Remark 3 and §3.3 to compute det(Λ^{#n})=det(Λ)^n and det(U)=1.
domain assumption The h-principle for totally real embeddings and Dimitroglou Rizell's Lagrangian approximation result turn the reversed decomposable concordance into a genuine Lagrangian concordance S(T^n).
Invoked in §3 without proof, citing [7], [8, §5], and [9, §2]; this is the main non-self-contained step.
domain assumption Sabloff–Vela-Vick–Wong Lemma 3.2: there is a genus-one Lagrangian cobordism from (S+S−)(Λ') to Λ'.
Used in §3.1 to increase genus by concatenation; cited to [16].
domain assumption Every decomposable Lagrangian cobordism between Legendrian knots in S^3_std is ribbon.
Used at the end of §3.3 to conclude non-decomposability from non-ribbonness; stated in §1 as a known fact.

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Pith. "Pith review of Non-decomposable Lagrangian cobordisms between Legendrian knots." pith.science (2026). https://pith.science/paper/Z7EJS726

@misc{pith2026251108731,
  author       = {Pith},
  title        = {Pith review of: Non-decomposable Lagrangian cobordisms between Legendrian knots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7EJS726}},
  note         = {Machine review of arXiv:2511.08731}
}

abstract

For a given $g>0$, we construct a family of non-decomposable Lagrangian cobordisms of genus $g$ between (stabilized) Legendrian knots in the standard contact three-sphere. The main technique we use to obstruct decomposability is based on Livingston's estimates.

Figures

Figures reproduced from arXiv: 2511.08731 by the authors.

**Figure 1.** Left: Lagrangian filling of the tb = −1 unknot; Right: Lagrangian pair-of-pants cobordism obtained from the Lagrangian 1-handle attachment. of the result of Cornwell-Ng-Sivek [6, Theorem 3.2], and in [9] we relied on the result of Agol which says that ribbon (stably homotopy ribbon) Lagrangian concordances of knots form a partial order, see [1, Theorem 1.1]. In this paper, we rely on the obstruction of ribbon cobord… view at source ↗

**Figure 2.** Decomposable Lagrangian concordance Ck from the tb = −1 Legendrian unknot U to the the Legendrian representative Λk of the pretzel knot P(3, −3, k) induced by the ambient surgery along the red arc. The front projection of Λk has k − 3 crossings in the blue box. Examples B. In order to describe the following family we first recall a few properties of determinant and Alexander polynomial. Remark 3. Determinant and A… view at source ↗

**Figure 3.** Lagrangian cobordism of genus one from (S+S−)(Λ′ ) to Λ′ obtained by making an ambient surgery along two red arcs. We apply this construction inductively to Λ′ = S(Λn ), in other words we concatenate the genus one Lagrangian cobordisms from [16, Lemma 3.2] in order for a given g > 0 to get a genus g Lagrangian cobordism from (S+S−) g (S(Λn )) to S(Λn ). We then concatenate the obtained cobordism and S(T n ) and get… view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

16 extracted references · 2 linked inside Pith

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Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [7]

Dimitroglou Rizell, Lagrangian approximation of totally real concordances , Proc

G. Dimitroglou Rizell, Lagrangian approximation of totally real concordances , Proc. London Math. Soc. (3) 2025;130:e70042

2025

[2] [3]

Borodzik and P

M. Borodzik and P. Tru¨ ol, Non-complex cobordisms between quasipositive knots , preprint 2025, available at 2504.04894

arXiv 2025

[3] [1]

Agol, Ribbon concordance is a partial ordering , Commun

I. Agol, Ribbon concordance is a partial ordering , Commun. Am. Math. Soc. 2 (2022), 374–379

2022

[4] [2]

Bae and I

Y. Bae and I. S. Lee, On Alexander polynomials of pretzel links , Kyungpook mathematical journal, 60(2):239–253, June 2020

2020

[5] [4]

Chantraine, Some non-collarable slices of Lagrangian surfaces , Bull

B. Chantraine, Some non-collarable slices of Lagrangian surfaces , Bull. Lond. Math. Soc. 44 (2012), no. 5, 981–987

2012

[6] [5]

Chantraine and N

B. Chantraine and N. Legout, Doubly slice knots and obstruction to Lagrangian concordan ce, Comptes Rendus. Math´ ematique, Volume 361 (2023), 1605–1609

2023

[7] [6]

Cornwell, L

C. Cornwell, L. Ng, and S. Sivek, Obstructions to Lagrangian concordance , Algebr. Geom. Topol. 16 (2016), no. 2, 797–824

2016

[8] [8]

Dimitroglou Rizell and R

G. Dimitroglou Rizell and R. Golovko, Instability of Legendrian knottedness, and non-regular La grangian concordances of knots , preprint 2024. Available at arXiv:2409.00290

arXiv 2024