{"id":"c1df1835-20df-4b93-b33f-5ed00413a22d","arxiv_id":"2507.15060","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Recursive Morse matchings compute integral Khovanov homology for all 4-strand torus links with at least 28 strands, including abundant 4-torsion and agreement with the Gorsky-Oblomkov-Rasmussen conjecture at infinity.","lead":"This paper defines two automatic Morse-style cancellation rules for Khovanov homology and uses them to compute integral Khovanov homology of 4-strand torus links, finding abundant 4-torsion. The computations confirm a known conjecture at the infinite limit and yield lower bounds on rational Gordian distance between torus knots.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1's omitted case is the load-bearing gap: the classification W=U is unproven for the g4∘IB∘g2∘IA family, and all recursions depend on it.","rationale":"The reader identified Lemma 5.1's omitted case as the weakest assumption, and I agree. This is the single most load-bearing concern because the classification of unmatched cells as formal words W is the foundation for Proposition 4.1, which in turn supplies the bijections an, bn, cn used in Theorem 1.1. The proof of Lemma 5.1 explicitly says 'We omit the details of the last case', and this is not a cosmetic omission: the family (g4 ∘ IB ∘ g2 ∘ IA)({e}) is one of the four components of W and is essential for the bn recursion. The only evidence for this case is code verification up to n = 50, which does not cover the infinite induction. If this classification fails at some larger n, the bijections of Proposition 4.1 would not be well-defined, and the recursions of Theorem 1.1 would not follow. I also considered the complexity of the cn differential correspondence in Section 6.2, which contains many diagrammatic and 'can be verified' assertions; however, those are gaps in the exposition of Proposition 4.4, whereas the omitted case in Lemma 5.1 is an explicit admission of an unproved statement in a lemma that everything downstream depends on. The paper has genuine strengths: reproducible scripts, computer verification of many base cases, and a novel algorithmic framework. No internal contradiction was found, and the Mgr counterexample concerns other braids, not the 4-strand torus braids. Therefore the appropriate verdict remains CONDITIONAL, as the reader stated; my analysis does not change that verdict, so the verdict should be UNCHANGED.","tokens_in":46172,"tokens_out":3685,"duration_ms":43984,"concrete_test":"Independently complete the omitted case of Lemma 5.1 by implementing Algorithm 4 for Mgr and symbolically verifying Equivalence 24 for all a ∈ (g4 ∘ IB ∘ g2 ∘ IA)({e}). Because the family is generated by periodic insertion of the B-block 101011000110, reduce to a finite collection of residue classes and verify the equivalence either by exhaustive enumeration of the finite-state automaton for n up to at least 200, or by a machine-checked proof in Lean that the periodicity implies the claim for all n. If the equivalence fails for any word in this family, the recursions and homology tables are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central recursions of Theorem 1.1 are derived from Proposition 4.1, whose proof rests entirely on Lemma 5.1 asserting that the unmatched cells U of the greedy Morse complex equal the formal word set W. The proof of Lemma 5.1 explicitly omits the final case a ∈ (g4 ∘ IB ∘ g2 ∘ IA)({e}) when verifying Equivalence 24, the inductive step from Un to Un+1. This family generates words containing the B-block 101011000110, which are used to define the bn correspondence and are present in unmatched cells for all sufficiently large n by Lemma 5.2. A failure of Equivalence 24 for any word in this family would invalidate W = U, hence Proposition 4.1, hence Theorem 1.1. The only backup given is computational verification for n ≤ 50, which does not cover the induction range n ≥ 83 used in Theorem 1.2. Moreover, Lemma 5.3 and Lemma 4.5 both rely on W = U through Lemma 5.2, so the entire combinatorial skeleton depends on this omitted case. The conditionality of the reader's verdict is therefore justified; the omission is not merely a presentational gap but a genuinely unproven step in the logical chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an algorithmic discrete Morse theory for Bar-Natan's tangle complexes, with cancellations performed globally after all Morse layers are processed. The main application is to negative 4-strand torus links T(4,-n): Theorem 1.1 gives two families of recursion isomorphisms for unreduced and reduced Khovanov homology, Theorem 1.2 uses these recursions plus computer base cases for n=28,...,82 to determine integral homology tables for all n>=28, and Section 4.3 compares the stable limit with the GOR conjecture. The paper also derives a splitting of Naot's Z[G]-complex for T(4,2n+1) and, via the lambda-invariant, lower bounds of 2 on rational Gordian distances from these torus knots. The proofs rest on a combinatorial classification of the unmatched cells of the greedy matching (W=U, Lemmas 5.1-5.3) and on a detailed path-correspondence argument for the an and cn recursions (Section 6).","tokens_in":46454,"tokens_out":9930,"duration_ms":108065,"significance":"Assuming the combinatorial claims can be completed, this is a significant advance. It would give the first complete integral Khovanov homology for an infinite family of 4-strand torus links, exhibiting infinitely many Z/4Z summands; it confirms the GOR conjecture over F2 in a large degree range, and the rational Gordian distance bounds are new geometric applications. The methodology is original and the paper has clear strengths: the main recursions are derived from a fixed matching rather than fitted to the answer, the computer-aided steps (Lean linarith, basecases.py, Khoca) are documented, and the comparison with the GOR conjecture is against an external prediction. The central claims are specific and falsifiable. However, the completeness of the paper currently depends on several unproven or under-proven combinatorial assertions, most importantly the omitted final case in Lemma 5.1.","major_comments":[{"comment":"The proof of W=U, the classification of unmatched cells, explicitly omits the final case 'showing that Equivalence 24 holds for a in (g4 o IB o g2 o IA)({e})'. This family generates exactly the B-block 101011000110 that defines the bn correspondence and is used in Lemma 5.2. Since Proposition 4.1, Proposition 4.4, and hence Theorem 1.1 all rely on W=U, this omission is load-bearing. The sentence reporting verification with braidalgo.py for n=0,...,50 does not close the gap: the induction in Theorem 1.2 proceeds from base cases n=28,...,82 to all n>=83, so the equivalence must be established for all word lengths. A complete proof of this case, or a certified exhaustive check for all n, is required.","section":"Section 5, Lemma 5.1"},{"comment":"The bound 'the longest word in W which does not contain A^2, B^2 or C^2 has length 3*23' is asserted with the justification 'by staring at the definition of W'. This is a nontrivial finiteness statement about the infinite set W and is the basis for the induction in Lemma 5.3, Lemma 4.5, and Proposition 4.1: it guarantees that every sufficiently long unmatched cell lies in the image of some gamma_*. Since the rest of the paper depends on this lemma, the assertion must be replaced by a verifiable argument, for example a regular-language or finite-state proof, or a machine-checked enumeration.","section":"Section 5, Lemma 5.2"},{"comment":"In Case II of the acyclicity proof, the statement 'By staring at the diagrams of a and x, one can conclude that isopair(y,j,k)=* ...' is not a proof. Proposition 3.6 is the result that Mgr is a Morse matching on Psi J(sigma1 sigma2 sigma3)^n K; it is used to define the complexes C_n and in the an-case of Proposition 4.4. The omitted check is finite and local, so it should either be written out explicitly or replaced by a small verified case analysis. The same comment applies to the 'straightforward evaluation of T' used in Section 6.2.2 to prove that Phi_n is well-defined.","section":"Section 3, Proposition 3.6"},{"comment":"The proof that the maps Lambda(B_{n-2}(p)) are orientation-preserving rests on the assertion that 'a straightforward evaluation of T (albeit tedious due to the number of cases)' gives T(p1, r_{2l+1}) >= 1 for all p in A_n. This inequality is the precise condition needed to apply Lemma 6.4 and Lemma 6.5, so it is load-bearing for the cn-case of Proposition 4.4. The evaluation should be presented or the cases should be mechanically checked, rather than left as an unstated computation.","section":"Section 6.2.2, Task 2"}],"minor_comments":[{"comment":"The proof is omitted as a 'case-by-case application of relations'; since this lemma classifies all isomorphisms used by the matching algorithms, the case analysis should be included or made available in an appendix.","section":"Section 2.2, Lemma 2.2"},{"comment":"The caption states 'For n >= 14' while Theorem 1.2 states that the results hold for n >= 28; please clarify whether the figure is valid for n >= 14 and adjust either the caption or the theorem statement accordingly.","section":"Figure 1 caption"},{"comment":"The passage 'Observing Figures 1 and 17 ... one can see' should be replaced by an explicit derivation of the Z/4Z summands in Kh(T(4, infinity)), since this is the evidence for agreement with the GOR conjecture in the stable limit.","section":"Section 4.3, Proposition 4.8"},{"comment":"The exceptional braid diagram data and the exhaustive search are only available through the repository [Kel25]; please include the relevant output as supplementary material or describe the search in sufficient detail to make the numerical result reproducible.","section":"Section 7, Numerical Result 7.1"}],"recommendation":"major_revision","confidential_remarks":"The repository [Kel25] is cited for essential verification scripts (basecases.py, braidalgo.py) and for base-case data in Theorem 1.2. I would ask the editor to ensure that this repository is publicly accessible and that the scripts are archived with the paper; the current manuscript's completeness depends on them. The omitted case in Lemma 5.1 is the substantive obstacle; if the author can supply a complete proof, the paper should be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper with a genuinely new result, but the central proof has an explicitly admitted gap and I would not take the main theorem on faith yet.\n\nWhat is new and good: the Mlex and Mgr matchings are new algorithmic tools; Mlex is proven to work on all tangle diagrams, and Mgr fails for exactly one braid in a 2977-diagram test set, which is honestly reported. The application is the first complete integral Khovanov homology for the 4-strand torus link family T(4,-n), including two recursions, lots of Z/4 torsion, agreement with the GOR conjecture in the stable limit, and rational Gordian distance bounds from the lambda-invariant. The paper ships code and data, and uses Khoca, Lean, and Python for base cases; that is real, reproducible evidence.\n\nThe soft spot is serious. Lemma 5.1, the classification W=U of unmatched cells, is load-bearing: Proposition 4.1, the recursions in Theorem 1.1, and the full induction in Theorem 1.2 all depend on it. The proof of Lemma 5.1 explicitly omits the last case, the family a in (g4 composed with IB composed with g2 composed with IA)({e}), and the only backup is a computer check for n up to 50. That does not cover the n >= 83 range used by the induction, so this is an unproven step in the logical chain, not a cosmetic omission. The stress-test note is right about this. The cn differential correspondence in Section 6.2 also leans on diagrams and 'straightforward evaluation' in several places; I do not see a fatal contradiction, and the strategy is plausible, but a referee will need to see that material fleshed out or machine-verified.\n\nSmaller quibbles: Lemma 5.2 is asserted 'by staring at the definition'—true, but it is a lemma in a paper of this length and should be written out or checked; and the numerical minimality comparison in Section 7 is a nice sanity check but not evidence for the theorem.\n\nWho this is for: people working on Khovanov homology of torus links, discrete Morse theory, or torsion in link homology. The result is important enough and the evidence strong enough that it deserves a serious referee. My recommendation: send to peer review, with the clear expectation of major revision—the omitted case in Lemma 5.1 has to be filled or independently verified.","headline":"A substantial new computation of Khovanov homology for 4-strand torus links, built on a proof with an explicitly admitted load-bearing gap in Lemma 5.1; worth refereeing, but not yet reliable as stated.","tokens_in":46966,"tokens_out":3564,"would_cite":false,"duration_ms":40301,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18","57K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves two periodicity recursions for the integral Khovanov homology of negative 4-strand torus links and, with computer-checked base cases, determines the homology tables for every n at least 28, exposing infinitely many…","keywords":["Khovanov homology","torus links","discrete Morse theory","Bar-Natan tangle complex","integral homology torsion","Z[G]-complexes","rational Gordian distance","greedy matching"],"falsifier":"Run the greedy-matching algorithm on the braid $(\\sigma_1\\sigma_2\\sigma_3)^{51}$ and compare the unmatched cells with the formal word set $W$; specifically, test Equivalence (24) for the omitted case $a \\in (g_4 \\circ I_B \\circ g_2 \\circ I_A)(\\{e\\})$. Any mismatch refutes $W=U$, and with it the recursion theorem, since all later arguments depend on that classification.","tokens_in":45971,"feed_emoji":"🧶","tokens_out":15742,"duration_ms":140365,"temperature":0.7,"pith_summary":"Khovanov homology is the bigraded knot invariant categorifying the Jones polynomial. For negative 4-strand torus links the paper proves the groups $\\mathrm{Kh}_{i,j}(T(4,-n))$ eventually repeat as $n$ grows: shifting $n$ by $4$ shifts the bidegree by $(0,12)$ in one grading region and by $(8,24)$ in another. The paper proves these recursions for unreduced and reduced Khovanov homology, and combines them with vanishing bounds and computer-verified base cases $n=28,\\ldots,82$ to determine the integral homology tables for all $n \\geq 28$. The stable limit $\\mathrm{Kh}(T(4,\\infty))$ then contains $\\mathbb{Z}/4\\mathbb{Z}$ torsion in bidegrees $(9+4k,14+6k)$ for every $k\\geq 0$, and over $\\mathbb{F}_2$ the computations agree with the Gorsky–Oblomkov–Rasmussen conjecture. A separate corollary splits a $\\mathbb{Z}[G]$-complex off the knot invariant, yielding lower bounds of at least $2$ on proper rational Gordian distances from $T(4,2n+1)$.","feed_headline":"Periodicity recursions pin down 4-strand torus link homology","feed_subtitle":"New Morse-theoretic cancellations give full integral tables and infinitely many order-four torsion groups.","key_machinery":"The engine is the greedy matching $M_{\\mathrm{gr}}$, a maximal collection of pairwise disjoint isomorphism edges in the graph of Bar-Natan's local tangle complex after delooping (replacing each circle by two quantum-shifted summands); a matching is Morse when reversing its edges leaves no directed cycles, and then the unmatched cells form a homotopy equivalent complex. For the 4-strand torus braid $(\\sigma_1\\sigma_2\\sigma_3)^n$, the unmatched cells are classified as formal words $W$ built from three periodic blocks $A=1^{12}$, $B=101011000110$, $C=(010001)^2$, and the maps $a_n,b_n,c_n$ duplicate the first occurrence of a block, lengthening the braid by 12 crossings. Proposition 4.4 shows these maps commute with the differentials in the appropriate grading regions, which is exactly what turns block duplication into the homology recursions of Theorem 1.1. Grading functions $t_A,t_B,t_C$ control when each block must appear, yielding the vanishing bounds needed for the induction over computer-verified base cases.","core_discovery":"The central claim is Theorem 1.1: for all $n\\geq 0$ and all gradings satisfying $-2n+2i-j\\geq 14$, both unreduced and reduced Khovanov homology satisfy $\\mathrm{Kh}_{i,j}(T(4,-n)) \\cong \\mathrm{Kh}_{i,j-12}(T(4,-n-4))$ (and the reduced analogue), and for $-9n+4i-3j\\geq 41$ they satisfy $\\mathrm{Kh}_{i,j}(T(4,-n)) \\cong \\mathrm{Kh}_{i-8,j-24}(T(4,-n-4))$. Because Proposition 4.6 gives vanishing outside two explicit half-planes, these recursions cover every bidegree once $n\\geq 28$, so the remaining job is a finite computer check of base cases; the paper performs it for $n=28,\\ldots,82$ and thereby obtains the full integral tables. In the stable limit, the tables produce 4-torsion exactly at the bidegrees predicted by the conjecture, and over $\\mathbb{F}_2$ the stable homology coincides with the Koszul-complex homology of the conjecture.","pith_inferences":["If the greedy matching is acyclic for every $m$-strand torus braid, the same block-duplication strategy should yield recursive descriptions for $m>4$; the missing ingredient is an analogue of the $b_n$ commutation proof.","The paper points to a third recursion that would describe $\\mathrm{Kh}_{i,j}(T(4,n))$ for all gradings and all $n$; proving it would remove the $n\\geq 28$ restriction and make the whole table a closed formula.","A small algorithmic proof for the omitted final case of Lemma 5.1 would eliminate the only computer-dependent step in the classification of unmatched cells, making the induction self-contained.","The $\\mathbb{Z}[G]$-splitting argument should transfer to any family whose reduced Khovanov homology is thin in high homological degrees, since only the top part is needed to recover the $G$-action."],"forward_implications":["For every negative 4-strand torus link with at least 28 crossings in the braid word, the full integral Khovanov homology table is now explicit, including free part and torsion.","The stable Khovanov homology $\\mathrm{Kh}(T(4,\\infty))$ contains infinitely many $\\mathbb{Z}/4\\mathbb{Z}$ summands, one in each bidegree $(9+4k,14+6k)$ for $k\\geq 0$.","Over $\\mathbb{F}_2$, the stable computations match the Gorsky–Oblomkov–Rasmussen conjecture in the region $i \\geq 42$, $j \\geq \\frac{3}{2}i - 1$, giving the first verification of the conjecture for 4 strands in this range.","A knot with at most $4n$ positive crossings cannot be changed into $T(4,2n+1)$ by a single proper rational tangle replacement; distinct odd 4-strand torus knots with $|n|,|m|\\geq 5$ also require at least two such replacements."],"supporting_citations":[{"why":"Defines Khovanov homology, the invariant whose integral groups are being computed.","marker":"[Kho00]"},{"why":"Constructs the local tangle complexes and delooping on which the greedy and lexicographic matchings act.","marker":"[Bar05]"},{"why":"Supplies algebraic discrete Morse theory, the cancellation theorem that justifies passing to the unmatched-cell complex.","marker":"[Sk¨o06]"},{"why":"Establishes the stable limit of torus-link Khovanov homology, the object compared with the GOR conjecture.","marker":"[Sto07]"},{"why":"Conjectures the stable Khovanov homology of torus knots as Koszul-complex homology; the paper verifies agreement over the two-element field and on 4-torsion.","marker":"[GOR13]"},{"why":"Provides the integer Khovanov homology computations used as induction base cases for n=28,...,82.","marker":"[LL16]"},{"why":"Introduces the Z[G]-complex formalism whose splitting produces the rational Gordian distance bounds.","marker":"[Nao06]"},{"why":"Introduces the graded lambda-invariant that turns the Z[G]-splitting into lower bounds on rational Gordian distance.","marker":"[LMZ24]"},{"why":"Supplies the code and data that verify the base cases and the numerical classification of unmatched cells.","marker":"[Kel25]"}],"fun_headline_variants":["4-torsion in all homological degrees for 4-strand torus links","New recursion computes full integral Khovanov homology of T(4,n)","Morse-theoretic cancellations give complete torus link homology tables","Stable 4-torsion matches Gorsky–Oblomkov–Rasmussen conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the classification $W=U$ of unmatched cells as formal words: the proof in Lemma 5.1 omits one case and falls back on computer verification for $n \\leq 50$, so if the greedy matching produces any unmatched cell outside $W$ for some $n > 50$, the recursions and all homology tables for large $n$ would have to be revised.","fun_headline_variants_meta":{"raw":{"variants":["4-torsion in all homological degrees for 4-strand torus links","New recursion computes full integral Khovanov homology of T(4,n)","Morse-theoretic cancellations give complete torus link homology tables","Stable 4-torsion matches Gorsky–Oblomkov–Rasmussen conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000335,"raw_usage":{"total_tokens":1867,"prompt_tokens":965,"completion_tokens":902,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":815}},"tokens_in":581,"tokens_out":902,"duration_ms":10033,"temperature":1.0,"reasoning_tokens":815,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:40:57.793575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the greedy-matching algorithm on the braid $(\\sigma_1\\sigma_2\\sigma_3)^{51}$ and compare the unmatched cells with the formal word set $W$; specifically, test Equivalence (24) for the omitted case $a \\in (g_4 \\circ I_B \\circ g_2 \\circ I_A)(\\{e\\})$. Any mismatch refutes $W=U$, and with it the recursion theorem, since all later arguments depend on that classification.","supporting_citations":[],"review_version":1}