{"id":"c3c7944e-6b53-4a5a-b2e2-3387f2b74958","arxiv_id":"2509.07921","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bordered LSFT algebras are defined for half-knots, and a pushout theorem recovers Ng's commutative LSFT algebra of the whole Legendrian knot.","lead":"A new construction attaches differential graded algebras to halves of a Legendrian knot diagram, and proves the commutative Legendrian Rational Symplectic Field Theory algebra of the whole knot arises as a pushout of these half-knot algebras. This extends Sivek's bordered Chekanov-Eliashberg construction to include disks with multiple positive punctures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1’s universal property silently requires the test DGA Q to be commutative; for arbitrary DGA targets the proposed h is not well-defined.","rationale":"The paper constructs explicit bordered versions of LSFT, supplies proofs of d^2=0 for the algebras (with some steps delegated to figures), and includes a detailed trefoil example and code. Those are meaningful checks of the construction. The central pushout theorem, however, has a genuine unstated hypothesis: the proof of the universal property defines h on the commutative quotient A^comm_SFT and extends it by h(xy)=h(x)h(y), which is only a valid definition if the images commute in Q. Since Q is allowed to be any DGA, this is not guaranteed. This is not a disagreement with a consensus; it is an internal gap in the argument. In the category of all DGAs, the pushout is the noncommutative amalgamated free product, so a commutative algebra can satisfy the universal property only if the relevant images commute in every test DGA, which is not true in general. The reader identified exactly this issue, and the proposed concrete test would make it decisive: if the trefoil left and right algebras have a nonzero commutator in their amalgamated free product, Theorem 5.1 is false as stated, and the theorem must be reformulated for commutative DGAs (or the images must be shown to commute). This is fixable but material, so the appropriate verdict remains CONDITIONAL, unchanged from the reader’s assessment.","tokens_in":32382,"tokens_out":7737,"duration_ms":79509,"concrete_test":"Test the claimed universal property against the actual pushout in the category of all DGAs: take Q = A^L_SFT ∗_{A^M_SFT} A^R_SFT, the noncommutative amalgamated free product, with canonical maps i_L and i_R. For the trefoil example of Section 6.1, check whether i_L(p_0)i_R(p_1) = i_R(p_1)i_L(p_0) holds in Q. Since p_0 and p_1 lie in different factors and are not identified through A^M_SFT, this commutator should be nonzero. If it is nonzero, no DGA map h:A^comm_SFT→Q can satisfy h∘L=i_L and h∘R=i_R, because A^comm_SFT is commutative; Theorem 5.1 is then false as stated. If, instead, all such commutators vanish, the proof must demonstrate this explicitly rather than assume it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5.7, the proof of Theorem 5.1 lets Q be any DGA and defines h:A^comm_SFT→Q by h(x)=f(s) if x=L(s), h(x)=g(s) if x=R(s), then extends by h(xy)=h(x)h(y). But A^comm_SFT is the free algebra modulo all commutators, so a homomorphism from it to an arbitrary associative DGA forces the images of every pair of generators to commute in Q. The paper never assumes Q is commutative and never proves that the images of f(A^L_SFT) and g(A^R_SFT) commute in Q. Without that, h is not well-defined: if x=L(s) and y=R(t), the relation xy=yx in A^comm_SFT forces f(s)g(t)=g(t)f(s), which need not hold in Q. This is the gap identified in the reader’s weakest assumption, and it is load-bearing because the pushout assertion is the paper’s main theorem. If the intended statement is the pushout in the category of commutative DGAs, the quantifier over Q must say so explicitly; as written, the universal property is not established. The noncommutative pushout of A^L_SFT and A^R_SFT over A^M_SFT is instead the amalgamated free product, and no commutative algebra can be that universal object unless the images of the two maps commute.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs bordered versions of Legendrian Rational Symplectic Field Theory (LSFT) for a simple Legendrian front cut into left and right halves by a vertical dividing line. It defines a commutative middle DGA A^M_SFT, left and right DGAs A^L_SFT and A^R_SFT, and proves that their differentials square to zero. It then defines DGA maps ℓ, r, L, R among these algebras and claims in Theorem 5.1 that the resulting square is a pushout whose apex is the commutative LSFT algebra A^comm_SFT. A two-sided variant is sketched in Section 5.8, and Section 6 works through the full example of the Legendrian trefoil, including explicit formulas for the algebras, differentials, and gluing maps.","tokens_in":32626,"tokens_out":8521,"duration_ms":84663,"significance":"If the main theorem is correct, it gives a Seifert–van Kampen type decomposition for the commutative LSFT algebra, extending Sivek's bordered Chekanov–Eliashberg theory to the rational/multi-puncture setting. The paper is constructive and explicit: the algebraic definitions are laid out in detail, the proofs of (d^M)^2=0 and (d^L)^2=0 are largely written out, and the trefoil example is fully computed, with code made available. These are real strengths. However, the universal property of the claimed pushout is stated for an arbitrary DGA target without a commutativity assumption, and the chain-map theorem for the maps L and R is justified by figure-based cancellation arguments rather than a complete algebraic pairing. Both issues are load-bearing for the central claim.","major_comments":[{"comment":"The universal property is claimed for 'another DGA Q' with no commutativity hypothesis. But A^comm_SFT is the quotient of the free algebra by all commutators, so a DGA map h:A^comm_SFT→Q forces the images of all generators to commute in Q. In particular, if x=L(s) and y=R(t), the relation xy=yx in A^comm_SFT would force f(s)g(t)=g(t)f(s) in Q, which is not automatic for an arbitrary associative DGA. The theorem is therefore not established as stated. The intended statement is presumably the pushout in the category of commutative DGAs, in which case Q should be required to be commutative and this should be stated explicitly. As written, the quantifier over Q makes the central pushout assertion unsupported.","section":"Section 5.7, proof of Theorem 5.1"},{"comment":"The proof that L and R are chain maps is only sketched at the points where it matters most. For x=α_ij the argument says that the relevant terms 'come in pairs which cancel, as depicted in Figure 20', and for x=β_ij it says that exceptional disks 'precisely cancel' (Figures 21, 22). Since d, d^L, and d^R are defined by sums over disks and string insertions, the equality d∘L=L∘d^L requires an explicit, globally defined involution on the set of contributing terms, including all boundary and dividing-line cases. This is not merely a presentational matter: Theorem 5.47 is used directly in the proof of the pushout property, so the central claim depends on this missing detail.","section":"Section 5.5.1, Theorem 5.47"},{"comment":"Lemma 5.41 asserts h_L→h_L=δ^L_str(h_L), which is used in the proof of Theorem 5.36 to prove (d^L)^2=0. The proof says that 'boundary' terms are 'depicted in Figure 17, and are seen to cancel pairwise', but no precise pairing is given. Because this lemma is another load-bearing point for the left algebra being a DGA, the argument should be expanded into an explicit enumeration of the cancellations, or at least a clearly stated bijection between summands.","section":"Section 5.4, Lemma 5.41"}],"minor_comments":[{"comment":"The two-sided pushout theorem is stated as a theorem but its proof is delegated to the sentence that the earlier proofs 'may be easily adapted'; please include a proof or explicitly label the statement as a sketch.","section":"Section 5.8, Theorem 5.55"},{"comment":"The set W is described as 'the set of elements of A^L_SFT which are representable as broken closed strings', but it is formally a set of words; this distinction should be made precise.","section":"Section 5.3.2, Definition 5.23"},{"comment":"The proof of well-definedness of the string differential refers to 'Figure 3.8 in [9]' and 'Figure 3.13 in [9]' without reproducing or explaining the relevant arguments; since the numbering is from another paper, a short self-contained statement would help.","section":"Section 4, proof of Proposition 4.8"},{"comment":"In the proof of Theorem 5.1, the case in which a generator of A^comm_SFT has representations both as L(s) and R(t) is handled for α and β generators, but the corresponding verification for t and t^{-1} is not written out; this should be made explicit.","section":"Section 5.7"}],"recommendation":"major_revision","confidential_remarks":"The central construction appears promising and the algebraic framework is largely explicit, but the universal property of Theorem 5.1 needs a corrected category statement, and the figure-based chain-map and cancellation arguments need to be made rigorous. These issues are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper constructs bordered Legendrian Rational SFT: differential graded algebras A^L_SFT, A^R_SFT, A^M_SFT, and a Seifert–van Kampen type pushout recovering Ng's commutative LSFT algebra. That is genuinely new, extending Sivek's bordered CE algebra to the multiple-positive-puncture setting. The definitions are explicit, the trefoil example is fully worked, and a code link is provided. The proofs of (d^M)^2=0 and (d^L)^2=0 are largely written out; the Jacobi and derivation identities are checked on generators. This is a serious piece of work, not a re-coordinatization.\n\nThe main soft spot is exactly the one the stress-test identifies. In the proof of Theorem 5.1, the paper takes an arbitrary DGA Q with maps f:A^L_SFT→Q and g:A^R_SFT→Q, defines h on generators of A^comm_SFT by h(x)=f(s) or g(s), and extends by h(xy)=h(x)h(y). But A^comm_SFT is the free commutative algebra on these generators, so h is an algebra map only if the images of f and g commute in Q. The paper never states that Q is commutative, nor proves the commutation. As written, the universal property is not established for general DGA targets. The fix is easy: either state the pushout in the category of commutative DGAs, or add the missing hypothesis. But without that, the main theorem is conditional. This is not a fatal flaw in the construction, just a quantifier error in the statement and proof.\n\nLesser concern: the proofs that L and R are chain maps (Theorem 5.47) and Lemma 5.41 (h^L→h^L = δ^L_str(h^L)) rely on figure-based cancellation arguments rather than full algebra. These are plausibly correct — the figures are convincing — but they would be stronger with detailed algebra or machine-checked verification. That is a minor-to-moderate concern, fixable by more exposition.\n\nThe citation pattern looks fine. The paper builds on Sivek and Ng and credits them properly. The example computations are reproducible in principle given the code, though no commit hash is pinned.\n\nThe paper deserves a serious referee. The construction is likely correct and useful for Legendrian contact homology and SFT, but the pushout theorem needs revision before it should be accepted. If the author fixes the category statement and tightens the figure-based proofs, this would be a solid contribution.","headline":"The paper builds a genuinely new bordered LSFT and a plausible pushout theorem, but the universal property as stated is not proven: it silently needs the target DGA to be commutative.","tokens_in":33175,"tokens_out":2792,"would_cite":true,"duration_ms":25831,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D42","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the commutative Legendrian Rational Symplectic Field Theory algebra of a simple Legendrian knot is the pushout of two bordered differential graded algebras obtained by cutting its front projection along a vertical…","keywords":["Legendrian knots","Rational Symplectic Field Theory","bordered differential graded algebra","pushout square","Chekanov-Eliashberg DGA","front projection","holomorphic disks with multiple positive punctures","string differential"],"falsifier":"Work in the trefoil example of Section 6 and let $Q$ be a noncommutative DGA with maps $f$ and $g$ agreeing on the middle algebra but with two left generators $\\alpha$ and $\\alpha'$ whose images in the commutative quotient commute, while $f(\\alpha)f(\\alpha') \\neq f(\\alpha')f(\\alpha)$ in $Q$. The alleged universal map $h$ would assign both orders of the product to the same element of $A^{comm}_{SFT}(\\Lambda)$, forcing the two values to be equal; a computation where they differ would contradict the pushout property.","tokens_in":32143,"feed_emoji":"✂️","tokens_out":14639,"duration_ms":121348,"temperature":0.7,"pith_summary":"This paper builds a bordered version of Legendrian Rational Symplectic Field Theory (LSFT). For a Legendrian knot whose front projection is simple, cutting the front along a generic vertical line produces three commutative differential graded algebras: one for the left half-knot, one for the right half-knot, and one for the dividing line. The main theorem (Theorem 5.1) claims that these three algebras form a pushout square whose apex is the commutative LSFT algebra of the whole knot, so the full invariant is determined by the two halves glued together over the cut. The construction extends the bordered Chekanov-Eliashberg algebra by allowing holomorphic disks with several positive punctures, which is exactly what LSFT adds to ordinary Legendrian contact homology. A two-sided analogue (Theorem 5.55) gives the same gluing statement for adjacent bordered pieces.","feed_headline":"Cut a knot in two; its rational SFT algebra is the pushout","feed_subtitle":"A pushout of two bordered algebras recovers the commutative Legendrian rational symplectic field theory of the cut knot.","key_machinery":"The load-bearing object is a family of commutative DGAs generated by geometric symbols attached to the cut. $A^L_{SFT}$ and $A^R_{SFT}$ are generated by $p_i, q_i$ for crossings and cusps in the half-diagram, by $\\alpha_{ij}$ recording half-disks whose boundary interval on the dividing line runs from $i$ to $j$, by $\\beta_{ij}$ recording strands that pair dividing-line points, and by $t, t^{-1}$. The middle algebra $A^M_{SFT}$ is generated by $\\alpha^L_{ij}, \\alpha^R_{ij}, \\beta^L_{ij}, \\beta^R_{ij}$, with Hamiltonian $h_M = \\sum_{i<j} \\alpha^L_{ij}\\alpha^R_{ij}$. In every case the differential is the sum of an SFT bracket $\\{h,\\cdot\\}$ and a string differential $\\delta_{str}$; the bracket satisfies a Jacobi identity, $\\delta_{str}$ is a derivation of the bracket, and the two quadratic terms cancel to give $d^2=0$. The maps $\\ell, r, L, R$ are defined by counting admissible disks and half-disks with corners at crossings and boundary intervals on $M$; commutativity of the square is checked generator by generator.","core_discovery":"The central discovery is a Seifert-van Kampen statement for commutative Legendrian SFT. Cutting a simple Legendrian front $\\Lambda$ along a vertical line $M$, one obtains $A^M_{SFT}(M)$, $A^L_{SFT}(\\Lambda_L)$ and $A^R_{SFT}(\\Lambda_R)$, and the paper proves that $A^{comm}_{SFT}(\\Lambda)$ is their pushout: any pair of DGA maps from the left and right bordered algebras that agree on the middle algebra factors through $A^{comm}_{SFT}(\\Lambda)$. This means the full LSFT invariant, whose differential counts disks with arbitrarily many positive punctures, can be assembled from counts that live in the half-diagrams and on the dividing line. The proof constructs explicit gluing maps $\\ell, r, L, R$ by counting admissible disks and half-disks, and it shows that the bordered differential, written as SFT bracket with a Hamiltonian plus a string differential, squares to zero.","pith_inferences":["The theorem is best read as a pushout in the category of commutative DGAs; for a noncommutative target the construction would need the extra assumption that the two side images commute.","The same gluing pattern suggests a general local-to-global principle for SFT-type invariants: whenever bordered algebras can be defined for a cut, the commutative invariant is determined by a pushout, making cutting-and-gluing the default computational strategy.","The noncommutative case might be recovered by a homotopy pushout or a curved structure, since the paper notes that the noncommutative maps are not morphisms in general.","Comparing the pushout output to a direct full-front computation of $A^{comm}_{SFT}$ for the trefoil would validate the gluing and reveal any hidden dependence on the choice of dividing line."],"forward_implications":["For a simple front, $A^{comm}_{SFT}(\\Lambda)$ is determined by the left and right half-diagrams together with the pairing data on the dividing line; no count of disks crossing the whole front is needed.","The bordered construction specializes to the bordered Chekanov-Eliashberg algebra: setting all $\\beta$ and $\\alpha^{R}$ generators to zero and deleting the $t^{\\pm1}$ terms recovers the commutative quotient of the earlier pushout theorem.","The two-sided LR algebra satisfies its own pushout theorem (Theorem 5.55), so adjacent bordered pieces can be glued one after another to compute the invariant of a knot assembled from tangle diagrams.","The universal property packages LSFT as a local-to-global invariant: DGA-valued invariants of the halves that agree on the middle algebra factor uniquely through $A^{comm}_{SFT}(\\Lambda)$.","The example calculations give a checkable blueprint: the trefoil computation shows how the middle, left, and right algebras and their maps are meant to fit together in practice."],"supporting_citations":[{"why":"Supplies the combinatorial differential graded algebra underlying Legendrian contact homology, including the admissible-disk differential and the proof that d^2=0.","marker":"[3]"},{"why":"Provides the Symplectic Field Theory setup and the moduli spaces of holomorphic curves with one positive and many negative punctures that the CE differential counts.","marker":"[5]"},{"why":"Gives the resolution of front projections and the front-projection admissible-disk calculus used throughout the bordered construction.","marker":"[8]"},{"why":"Defines Legendrian Rational SFT, its Hamiltonian, string differential, and the commutative algebra $A^{comm}_{SFT}$ that the pushout reconstructs.","marker":"[9]"},{"why":"Introduces the bordered Chekanov-Eliashberg DGAs and the pushout theorem that this paper extends to multiple positive punctures.","marker":"[11]"},{"why":"Supplies the Legendrian Reidemeister moves used to arrange a front into the simple form required by the theorem.","marker":"[12]"}],"fun_headline_variants":["LSFT algebra of a knot is pushout of its halves","Seifert-van Kampen holds for Legendrian rational SFT","Bordered LSFT: glue half-knot algebras via pushout","Rational SFT from halves: a pushout theorem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The pushout theorem assumes the target algebra is commutative: the universal map is defined on the commutative quotient of the LSFT algebra, and the proof does not show it descends when the target is not commutative.","fun_headline_variants_meta":{"raw":{"variants":["LSFT algebra of a knot is pushout of its halves","Seifert-van Kampen holds for Legendrian rational SFT","Bordered LSFT: glue half-knot algebras via pushout","Rational SFT from halves: a pushout theorem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":2003,"prompt_tokens":822,"completion_tokens":1181,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":1109}},"tokens_in":438,"tokens_out":1181,"duration_ms":8423,"temperature":1.0,"reasoning_tokens":1109,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:09:17.341596+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work in the trefoil example of Section 6 and let $Q$ be a noncommutative DGA with maps $f$ and $g$ agreeing on the middle algebra but with two left generators $\\alpha$ and $\\alpha'$ whose images in the commutative quotient commute, while $f(\\alpha)f(\\alpha') \\neq f(\\alpha')f(\\alpha)$ in $Q$. The alleged universal map $h$ would assign both orders of the product to the same element of $A^{comm}_{SFT}(\\Lambda)$, forcing the two values to be equal; a computation where they differ would contradict the pushout property.","supporting_citations":[{"cited_title":"Differential algebra of Legendrian links.Inventiones mathematicae, 150(3):441–483, 2002","cited_arxiv_id":null,"evidence_quote":"Supplies the combinatorial differential graded algebra underlying Legendrian contact homology, including the admissible-disk differential and the proof that d^2=0."},{"cited_title":"Introduction to Symplectic Field Theory","cited_arxiv_id":null,"evidence_quote":"Provides the Symplectic Field Theory setup and the moduli spaces of holomorphic curves with one positive and many negative punctures that the CE differential counts."},{"cited_title":"Computable Legendrian invariants.Topology, 42(1):55–82, 2003","cited_arxiv_id":null,"evidence_quote":"Gives the resolution of front projections and the front-projection admissible-disk calculus used throughout the bordered construction."},{"cited_title":"Rational symplectic field theory for Legendrian knots.Inventiones mathemati- cae, 182(3):451–512, 2010","cited_arxiv_id":null,"evidence_quote":"Defines Legendrian Rational SFT, its Hamiltonian, string differential, and the commutative algebra $A^{comm}_{SFT}$ that the pushout reconstructs."},{"cited_title":"A bordered Chekanov–Eliashberg algebra.Journal of Topology, 4(1):73–104, 2011","cited_arxiv_id":null,"evidence_quote":"Introduces the bordered Chekanov-Eliashberg DGAs and the pushout theorem that this paper extends to multiple positive punctures."},{"cited_title":"On the isotopy of Legendrian knots.Annals of Global Analysis and Ge- ometry, 10(3):195–207, 1992","cited_arxiv_id":null,"evidence_quote":"Supplies the Legendrian Reidemeister moves used to arrange a front into the simple form required by the theorem."}],"review_version":2}