{"id":"b10e1634-def4-4ec3-ae45-4798ef1af3d4","arxiv_id":"2511.08731","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every genus g>0, the authors build non-decomposable Lagrangian cobordisms of genus g between stabilized Legendrian knots, using Livingston's estimates to obstruct decomposability.","lead":"This paper constructs Lagrangian surfaces of any positive genus that connect two Legendrian knots in the three-sphere but cannot be built from the standard elementary pieces. It gives the first examples of non-decomposable Lagrangian cobordisms with positive genus, using knot-determinant estimates to rule out decomposability.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The existence of the stabilized Lagrangian concordance S(T^n) in §3 is the load-bearing external premise; it is only cited to [7,8,9] and its hypotheses are not checked.","rationale":"I read the paper in good faith. Theorem 1 is clearly stated, and the obstruction argument after the existence of L_g is sound: a Lagrangian cobordism with c2>0 cannot be decomposable because decomposable implies ribbon, and the estimates (3.1)–(3.3) with p|det(Λ) give c2≥n/2−g. Proposition 6 follows from the behavior of H1 of double branched covers under connected sums. The determinant computations and the pretzel examples are consistent. The single point where the argument stops being self-contained is Section 3's construction of S(T^n). It is not a matter of consensus but of correctness risk: the authors explicitly say 'we argue the same way as in [9, Section 2] and [8, Section 5]', and both are same-author preprints; [7] is published but no theorem number or hypothesis check is supplied. Because the entire existence of a Lagrangian cobordism between the required stabilized ends rests on this step, it is the weakest load-bearing assumption. A reader who distrusts [7] or the preprint argument has no way to verify the construction from this paper alone. This warrants the CONDITIONAL verdict already given by the reader; no adjustment is needed. The proposed test—an independent re-derivation of S(T^n) against [7]—would settle whether the concern lands.","tokens_in":5626,"tokens_out":30251,"duration_ms":278385,"concrete_test":"Independently re-derive the passage from (C^n)^{-1} to S(T^n) following [8, §5] and [9, §2], and check the precise statement of the approximation theorem in [7] (e.g., Theorem 1.1 or its corollary): (i) T^n is cylindrical with Legendrian ends; (ii) its tangent planes define a formal Lagrangian Gauss map; (iii) the number and sign of stabilizations in S(·) meet the theorem's bound. If any hypothesis fails—e.g., if the h-principle output is only C^0-close to a formal Lagrangian rather than formally Lagrangian—then S(T^n) need not exist and Theorem 1 is unsupported. If all hypotheses hold, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction in §3 depends on the existence of the stabilized Lagrangian concordance S(T^n) from S(Λ^n) to S(U^n). That existence is not proved in this paper: the authors write '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 theorem from [7] is given, and no verification that the totally real concordance T^n obtained from the reversed decomposable concordance satisfies its hypotheses (cylindrical ends, formal Lagrangian condition, stabilization threshold) is included. If this step fails, the cobordism L_g is never produced, and the Livingston/Borodzik–Tröl estimate is never applied. The rest of the proof—determinant multiplicativity, Proposition 6, and the inequality c2≥n/2−g>0—is standard and appears correct. The weakness is therefore a single external dependency on unpublished/same-author preprints [8,9] and a strong theorem [7] whose applicability is not explicitly checked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5860,"tokens_out":13894,"duration_ms":146244,"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":[{"comment":"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.","section":"§3, paragraph beginning 'Then we argue the same way as in [9, Section 2] and [8, Section 5]'"},{"comment":"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.","section":"§3.3 / Theorem 5"}],"minor_comments":[{"comment":"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.","section":"§2, Examples C"},{"comment":"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.","section":"Theorem 1"},{"comment":"Minor wording: 'Formulas 3.1, 3.2 and 3.3' should refer to equations (3.1)–(3.3).","section":"§3.2"},{"comment":"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.","section":"§3"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central existence step for S(T^n) is the only serious obstacle; if the authors can supply a self-contained lemma or a precise verifiable citation, the result should be accepted. I would also ask the editor to ensure that the preprints [8] and [9] are publicly available or included as appendices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWhere 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.\n\nMinor 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.\n\nBottom 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.\n\nRecommendation: send to peer review, with a request to check the external dependencies.","headline":"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.","tokens_in":6391,"tokens_out":3522,"would_cite":true,"duration_ms":38139,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D12","53D42"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every genus, non-decomposable Lagrangian cobordisms exist","keywords":["Legendrian knots","Lagrangian cobordisms","decomposable cobordisms","ribbon cobordisms","two-fold branched covers","critical point estimates","determinant","stabilizations"],"falsifier":"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.","tokens_in":5464,"feed_emoji":"🔗","tokens_out":11337,"duration_ms":94714,"temperature":0.7,"pith_summary":"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.","feed_headline":"For every genus, non-decomposable Lagrangian cobordisms exist","feed_subtitle":"A branched-cover homology estimate forces an extra critical point, so these cobordisms cannot be built from births and surgeries.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Every genus yields non-decomposable Lagrangian cobordisms","Non-decomposable Lagrangian cobordisms for every genus","Homology estimate forces non-decomposable genus-g cobordisms","Genus-g Lagrangian cobordisms that resist decomposition exist"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every genus yields non-decomposable Lagrangian cobordisms","Non-decomposable Lagrangian cobordisms for every genus","Homology estimate forces non-decomposable genus-g cobordisms","Genus-g Lagrangian cobordisms that resist decomposition exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000684,"raw_usage":{"total_tokens":2864,"prompt_tokens":589,"completion_tokens":2275,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":333,"completion_tokens_details":{"reasoning_tokens":2219}},"tokens_in":333,"tokens_out":2275,"duration_ms":16976,"temperature":1.0,"reasoning_tokens":2219,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:47:04.556327+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}