{"id":"88fc1ddd-332b-4a0b-a8b2-ec121fe8b05e","arxiv_id":"2506.14050","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Separating half Giroux torsion does not force the contact invariant to vanish: infinitely many closed counterexamples exist, disproving Ghiggini's conjecture.","lead":"This paper constructs infinitely many closed contact 3-manifolds that contain half Giroux torsion along a separating torus while their contact invariants remain non-zero. The construction refutes Ghiggini's conjecture and shows that the minimal twisting needed to force the contact invariant to vanish is exactly 2π.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-vanishing of c(ξ) rests on manual bigon counts in Figures 4 and 6 that do not rule out differentials from generators outside the drawn local picture into c(ξ); without an algebraic or computer check the counterexample claim is not established.","rationale":"The reader's weakest assumption already identifies the manual bigon computations as the load-bearing point. My reading confirms this: the logical structure of the construction is coherent, and the cited results on innermost contact structures supply nonvanishing sutured invariants, but the final pairing is the step that must be checked. I do not see an internal inconsistency; the concern is lack of verification of a critical computation. Therefore the appropriate verdict is unchanged (CONDITIONAL), pending a full algebraic check of Figures 4 and 6.","tokens_in":12124,"tokens_out":4064,"duration_ms":47512,"concrete_test":"Recompute the two bordered pairings algebraically, using the explicit type-A modules for positive torus knot complements (from [19, Ch.11]) and the simplified type-D modules in Figures 3c/5c together with the torus-algebra differentials in [25, §4.1], for a spread of cases such as (p,q) = (-2,5), (-3,4), (-3,5) and the figure-eight double. Enumerate all generators and all A∞/tensor differentials and check whether any arrow has target c(ξ) besides the drawn ∂x = y + c(ξ). If no extra arrow is found in these cases, rerun for a larger sample or prove a grading bound excluding such arrows; if an extra arrow is found, Figures 4/6 are incomplete and the nonvanishing theorems are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.2 (and similarly Theorem 1.4) is that the contact class c(ξ) = cA(ξ123_{p,q}) ⊠ cD(ξin_{-2,3}) is nonzero in ˆHF(-Y_{p,q}). The proof in §3.2 identifies three generators x, y, c(ξ) in Figure 4, asserts an immersed bigon from x to y and from x to c(ξ), and concludes from 'these are the only bigons involving these three generators' that the three form a summand with ∂x = y + c(ξ), so c(ξ) survives. The gap: nonvanishing requires that no generator outside this triple has a differential hitting c(ξ). For a general negative torus knot T_{p,q}, the bordered complex contains many generators (roughly (pq)/2 Alexander gradings), and Figure 4 shows only a local fragment of the immersed curve. No algebraic computation, grading argument, or computer verification is given to rule out an additional bigon from some other generator z to c(ξ); if one existed, ∂z = c(ξ) and the invariant would vanish, destroying the counterexample. The same issue affects Figure 6 for the double of the figure-eight complement, where only generators x, y, z and c(ξ) are drawn. Since the counterexamples to Ghiggini's conjecture depend entirely on this nonvanishing, the proof is conditional on completing and checking these bigon counts.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit closed contact 3-manifolds that contain half Giroux torsion (respectively convex torsion) along a separating torus and whose Heegaard Floer contact invariants are claimed to be nonzero. Theorem 1.2 gives infinitely many such examples, obtained as splices of the left-handed trefoil complement with negative torus knot complements; Theorem 1.4 gives a single example, the double of the figure-eight knot complement, with convex torsion along a separating torus. The proofs use the authors' bordered contact invariant machinery from [25] together with innermost contact structures on knot complements, and the main nonvanishing computations are presented via immersed curve bigon counts.","tokens_in":12446,"tokens_out":6175,"duration_ms":65499,"significance":"If the main theorems are correct, they provide the first closed-manifold counterexamples to Ghiggini's conjecture and show that separating half Giroux torsion does not force the contact invariant to vanish. The paper also sharpens the known vanishing threshold for twisting along a separating torus. The use of bordered contact invariants is well matched to the construction, and the reduction of the convex torsion case to Giroux torsion via Proposition 2.5 is elegant. However, the central nonvanishing claims rest on manual bigon-counting arguments in two figures, and the manuscript does not supply the algebraic or machine-checked verification needed to rule out additional differentials into the contact class. This gap is load-bearing for both theorems.","major_comments":[{"comment":"The proof of nonvanishing of c(ξ) asserts that 'these are the only bigons involving these three generators' in a local fragment of the immersed curve diagram. This only controls the subcomplex generated by x, y, and c(ξ); a bigon from any intersection point outside the drawn fragment into c(ξ) would make the contact class exact, destroying the counterexample in Theorem 1.2. No algebraic computation, Maslov index bound, or computer verification is provided to rule out such bigons. This is a load-bearing gap and must be closed.","section":"Section 3.2, Figure 4"},{"comment":"The same issue affects Theorem 1.4: the four drawn generators are asserted to form a summand, but bigons from generators outside the local picture into c(ξ) are not excluded. Additionally, the statement that the vertical type-A curve has been homotoped 'for admissibility reasons' needs justification that this homotopy does not create or destroy bigons involving c(ξ). A complete differential computation or a formal grading argument is required to establish the claimed nonvanishing.","section":"Section 3.4, Figure 6"},{"comment":"The step from Proposition 2.13(4) to 'we know m2(cA(ξin_8),ρ1)≠0' is not justified as written, since ρ1 is described in Theorem 2.10 as a basic slice, not a half convex torsion layer; the surrounding argument uses ρ23. The identification of cA(ξin_8) as either E or E+I depends on this step, and if cA(ξin_8)=I were possible then the computation in Theorem 1.4 would collapse. Please clarify which generator corresponds to the half convex torsion layer and verify the resulting identification of cA(ξin_8).","section":"Section 3.3"}],"minor_comments":[{"comment":"The reference '[21, Propostion 2.5]' contains a typo: 'Propostion' should be 'Proposition'.","section":"Introduction, Question 1.10 paragraph"},{"comment":"The phrase 'bordered paring computation' should read 'bordered pairing computation'.","section":"Section 3.2"},{"comment":"The statement allows all coprime (p,q) with pq<0, but the proof fixes p<0<q; the case p>0>q should be addressed explicitly, for instance by symmetry of torus knots.","section":"Theorem 1.2"},{"comment":"The sentence 'Since B and T are the unique generators in Alexander gradings 1 and -1, respectively, they represent the contact invariants of conjugate contact structures' is terse; a brief justification or reference would help the reader.","section":"Section 3.3"},{"comment":"The phrase 'the minimal amount of twisting along a separating torus necessary to ensure that the contact invariant vanishes' could be misread as a universal statement; consider rephrasing to make clear that it means no smaller amount forces vanishing in general, since Theorem 1.4 provides one counterexample.","section":"Corollary 1.5"}],"recommendation":"major_revision","confidential_remarks":"The main constructions are interesting and the paper is well aligned with the journal's scope. The decisive issue is the manual bigon count in Figures 4 and 6; if the authors can supply a fully algebraic or computer-verified computation, or a grading argument that rigorously excludes bigons into the contact class, the paper would be a strong contribution. I do not see evidence of circularity, as the cited previous work is prior and independent of the new nonvanishing claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper almost certainly gives an infinite family of counterexamples to Ghiggini's conjecture, and the construction is genuinely new. But the proof of the key non-vanishing claim is not fully written down. The authors draw two figures, assert that the only bigons connecting certain generators are the ones they drew, and conclude that the contact class survives. A referee is going to want to see that this assertion is actually true for the whole complex, not just for the three generators in the local picture.\n\nWhat's new: the splice construction using innermost contact structures on negative torus knot complements, paired with the authors' bordered contact invariants, is a nice idea. It produces infinitely many closed manifolds with separating half Giroux torsion and non-vanishing contact invariant, which directly contradicts a conjecture that has been open for a while. The convex torsion example and the corollary that 2π is the minimal twisting threshold are also interesting, and the corollary about minimal twisting is a clean consequence.\n\nThe soft spots: the non-vanishing check in Figures 4 and 6 is the load-bearing step. The stress-test note is right: to prove that c(ξ) is not killed by a differential, you need to rule out bigons from all other generators, and the text only says 'these are the only bigons involving these three generators.' That's not the same as a summand. Maybe the local picture is the whole story because the curves are simple and the intersections are ordered, but the paper doesn't give that argument. This is fixable, but it needs to be fixed, not just asserted.\n\nThere's also a smaller issue in Section 3.3: the text says Proposition 2.13(4) implies m2(cA(ξin_8), ρ1) ≠ 0, but ρ1 is a basic slice, not a half convex torsion layer, according to their own Theorem 2.10. That looks like a typo, but it should be corrected because the logic of that paragraph depends on it.\n\nOverall, the paper is worth serious refereeing. The result is significant, the tools are appropriate, and the gaps are addressable. I'd send it to a good referee who can check the immersed curve computations, with a request that the authors either prove the global bigon claim or include a computer verification.\n\nRecommendation: accept the paper for peer review. Reading group: maybe, if someone wants to understand the bordered contact invariant machinery.","headline":"A likely disproof of Ghiggini's conjecture with a new construction, but the key non-vanishing computation is asserted by inspection rather than proved.","tokens_in":12919,"tokens_out":10863,"would_cite":true,"duration_ms":99095,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K33","57R17","57K18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs infinitely many closed contact 3-manifolds with half Giroux torsion along a separating torus whose contact invariants do not vanish, falsifying the conjecture stated as Conjecture 1.1 in its introduction.","keywords":["contact invariant","half Giroux torsion","bordered Floer homology","innermost contact structure","Ghiggini's conjecture (Conjecture 1.1)","immersed curves","symplectic fillability","convex torsion"],"falsifier":"An independent computation of the bordered pairing complex for Y_{p,q} (or the double of the figure-eight knot complement) that finds an immersed bigon not shown in Figure 4 (or Figure 6) among the generators x, y, c(ξ) (or x, y, z, c(ξ)) would alter the differential and could make the claimed contact class vanish in homology.","tokens_in":11935,"feed_emoji":"🌀","tokens_out":6227,"duration_ms":57925,"temperature":0.7,"pith_summary":"The paper tries to establish that half Giroux torsion along a separating torus does not force the Heegaard Floer contact invariant to vanish. It builds infinitely many closed contact 3-manifolds where the invariant is demonstrably non-zero, thereby contradicting Conjecture 1.1, which asserted the opposite for all closed manifolds. It also shows that a closed contact manifold with convex torsion along a separating torus can have a non-vanishing contact invariant, pinning down 2π as the minimal twisting amount that guarantees vanishing. The construction pairs innermost contact structures on knot complements with a torsion layer, and the non-vanishing is verified through bordered contact invariants and immersed-curve bigon counts.","feed_headline":"Half Giroux torsion does not kill contact invariants","feed_subtitle":"Infinite family of spliced trefoil–torus knot manifolds disproves the vanishing conjecture","key_machinery":"The argument is carried by the bordered contact invariants $c_A$ and $c_D$ developed by the authors: $c_A(\\xi)$ lives in a type-A module associated to a bordered sutured 3-manifold, and a pairing theorem (Theorem 2.8) recovers the sutured contact invariant of a glued manifold as $c_A(\\xi_1)\\boxtimes c_D(\\xi_2)$. The examples are built from innermost contact structures, the minimal tight contact structures on knot complements (defined by the non-thickening condition on the convex boundary), together with the element $\\rho_{123}$ in the punctured-torus algebra, which corresponds to a $3\\pi/2$-layer containing half Giroux torsion. The non-vanishing of $\\hat{c}(\\xi)$ is established by computing the placement $c_A(\\xi^{123}_{p,q}) \\boxtimes c_D(\\xi^{\\mathrm{in}}_{-2,3})$ through immersed curves in the punctured torus, where the differential counts immersed bigons; the bigons displayed in Figures 4 and 6 show that the class $c(\\xi)$ survives in homology.","core_discovery":"The paper's central claim is that, for any coprime integers (p,q) with pq<0, the splice Y_{p,q} of the left-handed trefoil and the (p,q)-torus knot admits a contact structure ξ_{p,q} that contains half Giroux torsion along a separating torus and whose contact invariant $\\hat{c}(\\xi_{p,q})$ is non-zero. If correct, this gives infinitely many closed-manifold counterexamples to Conjecture 1.1, which predicted that separating half Giroux torsion always kills the contact invariant. The paper also claims a contact structure on the double of the figure-eight knot complement that contains convex torsion along a separating torus and has non-vanishing contact invariant, implying that a full 2π twist is the minimal amount of twisting needed to ensure vanishing.","pith_inferences":["The paper's method gives a template for producing contact manifolds with unusual non-vanishing invariants: splice knot complements whose innermost contact structures have known bordered invariants, then insert a torsion layer whose algebra element is detectable under the pairing.","If the spliced manifolds Y_{p,q} turn out to be symplectically fillable, half Giroux torsion would join half convex torsion as a twist that fillability can tolerate, sharpening the distinction between torsion and convex torsion.","The immersed-curve bigon counts in Figures 4 and 6 are concrete enough to be checked by an automated search; such a verification would either confirm the theorem as stated or identify a missing differential, making this an unusually testable proof.","By analogy with the figure-eight case, doubling complements of even twist knots may yield an infinite family of closed manifolds with separating convex torsion and non-vanishing contact invariants, as the authors suggest in their concluding questions."],"forward_implications":["Conjecture 1.1 is false: separating half Giroux torsion does not force the contact invariant to vanish in closed manifolds.","Separating half Giroux torsion alone cannot obstruct symplectic fillability, because these examples carry the non-vanishing contact invariant that filling obstructions detect.","Convex torsion and Giroux torsion are inequivalent notions: a convex torsion layer can be present where no Giroux torsion layer embeds, and the contact invariant can survive the former but not the latter.","The minimal twisting amount along a separating torus that guarantees vanishing of the contact invariant is exactly 2π.","The same construction applied to splices of the left-handed trefoil with any negative L-space knot, whose complements admit innermost contact structures, would yield further counterexamples."],"supporting_citations":[{"why":"Supplies the bordered contact invariants c_A and c_D, the type-A/type-D pairing theorem (Theorem 2.8), and the correspondence between algebra elements and tight contact structures on T^2 x [0,1] used to attach torsion layers.","marker":"[25]"},{"why":"Establishes that the complement of every negative torus knot admits an innermost contact structure with non-vanishing sutured invariant (Proposition 2.14), the key input for the torus knot pieces and the left-handed trefoil piece.","marker":"[8]"},{"why":"Gives the innermost contact structure on the figure-eight knot complement with non-vanishing sutured invariant (Proposition 2.13), which underlies the convex-torsion example in Theorem 1.4.","marker":"[3]"},{"why":"Provides the immersed-curve interpretation of bordered Floer homology in the punctured torus that the bigon-counting computations in Figures 4 and 6 rely on.","marker":"[14]"},{"why":"Supplies the bordered modules of torus knot complements from knot Floer complexes, which determine the type-A module used for the torus knot piece.","marker":"[19]"},{"why":"Proves the contact invariant vanishes in the presence of Giroux torsion, the baseline vanishing theorem that the new half-torsion examples must evade.","marker":"[12]"},{"why":"Also proves vanishing of the contact invariant under Giroux torsion, providing the standard obstruction that Conjecture 1.1 sought to extend to half torsion.","marker":"[20]"}],"fun_headline_variants":["Half Giroux torsion fails to kill contact invariants","Counterexamples to Ghiggini conjecture from bordered invariants","Infinite family: half torsion with non-vanishing contact invariant","Convex torsion does not vanish contact invariant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The non-vanishing proof rests on the manual immersed-curve counts in Figures 4 and 6, which assume that the displayed bigons are the only differentials among the relevant generators; an extra bigon or differential could make the class c(ξ) null-homologous.","fun_headline_variants_meta":{"raw":{"variants":["Half Giroux torsion fails to kill contact invariants","Counterexamples to Ghiggini conjecture from bordered invariants","Infinite family: half torsion with non-vanishing contact invariant","Convex torsion does not vanish contact invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000897,"raw_usage":{"total_tokens":3795,"prompt_tokens":810,"completion_tokens":2985,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":426,"completion_tokens_details":{"reasoning_tokens":2920}},"tokens_in":426,"tokens_out":2985,"duration_ms":23871,"temperature":1.0,"reasoning_tokens":2920,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:18:08.600085+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent computation of the bordered pairing complex for Y_{p,q} (or the double of the figure-eight knot complement) that finds an immersed bigon not shown in Figure 4 (or Figure 6) among the generators x, y, c(ξ) (or x, y, z, c(ξ)) would alter the differential and could make the claimed contact class vanish in homology.","supporting_citations":[{"cited_title":"On contact invariants in bordered Floer homology","cited_arxiv_id":"2410.05511","evidence_quote":"Supplies the bordered contact invariants c_A and c_D, the type-A/type-D pairing theorem (Theorem 2.8), and the correspondence between algebra elements and tight contact structures on T^2 x [0,1] used to attach torsion layers."},{"cited_title":"Classification of tight contact structures on surgeries on the figure- eight knot","cited_arxiv_id":null,"evidence_quote":"Gives the innermost contact structure on the figure-eight knot complement with non-vanishing sutured invariant (Proposition 2.13), which underlies the convex-torsion example in Theorem 1.4."},{"cited_title":"Bordered Floer homology for manifolds with torus boundary via immersed curves","cited_arxiv_id":null,"evidence_quote":"Provides the immersed-curve interpretation of bordered Floer homology in the punctured torus that the bigon-counting computations in Figures 4 and 6 rely on."},{"cited_title":"Ozsvath, and Dylan P","cited_arxiv_id":null,"evidence_quote":"Supplies the bordered modules of torus knot complements from knot Floer complexes, which determine the type-A module used for the torus knot piece."},{"cited_title":"The vanishing of the contact invariant in the presence of torsion","cited_arxiv_id":"0706.1602","evidence_quote":"Proves the contact invariant vanishes in the presence of Giroux torsion, the baseline vanishing theorem that the new half-torsion examples must evade."},{"cited_title":"Stipsicz","cited_arxiv_id":null,"evidence_quote":"Also proves vanishing of the contact invariant under Giroux torsion, providing the standard obstruction that Conjecture 1.1 sought to extend to half torsion."}],"review_version":1}