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REVIEW 4 major objections 4 minor 10 references

Morse matchings and Khovanov homology of 4-strand torus links

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.15060 v1 pith:ASRFQETE submitted 2025-07-20 math.GT math.CO

Tuomas Kelomäki This is my paper · ORCID
classification math.GTmath.CO MSC 57K1857K10
keywords KhovanovhomologytoruslinksdiscreteMorsetheoryBar-NatantanglecomplexintegraltorsionZ[G]-complexesrationalGordiandistancegreedymatching
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

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 $T(4,2n+1)$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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).

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 (4)
  1. [Section 5, Lemma 5.1] 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.
  2. [Section 5, Lemma 5.2] 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.
  3. [Section 3, Proposition 3.6] 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.
  4. [Section 6.2.2, Task 2] 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.
minor comments (4)
  1. [Section 2.2, Lemma 2.2] 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.
  2. [Figure 1 caption] 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.
  3. [Section 4.3, Proposition 4.8] 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.
  4. [Section 7, Numerical Result 7.1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction in the Khovanov recursion derivation; the load-bearing gap is the omitted final case of Lemma 5.1, which is a completeness risk, not a circularity.

full rationale

Score 1: essentially self-contained, non-circular derivation with two flagged caveats. The main recursions (Theorem 1.1) are derived, not fit: Proposition 4.1 defines the bijections a_n, b_n, c_n combinatorially by gamma_A, gamma_B, gamma_C (replacing the first occurrence of a fixed length-12 word by its square), and Proposition 4.4 proves differential commutation by explicit path correspondences — a graph embedding with sign bookkeeping (Lemma 6.1) for a_n, and the Phi_n surgery map with R-equivalence classes (Section 6.2) for c_n. No parameter is tuned to the target homology: the grading shifts (0,12) and (8,24) are read off from the grading effect of gamma_* in the proof of Proposition 4.1, and the inequality regions of Theorem 1.1 come from the fixed linear functions t_A, t_B, t_C. Theorem 1.2 rests on concrete base cases n = 28,...,82 computed by the external program Khoca [LL16] and checked by Python scripts, so the induction is grounded outside the paper's own machinery. The GOR comparison (Propositions 4.8 and 4.9) is against an external conjecture with its own Koszul/Poincare series; Proposition 4.8 proves the algebraic 4-torsion by a self-contained ideal-membership argument (Equation 21), and the topological side is read off from the independently computed tables. The only self-citation is Section 3, where [Kel24] supplies acyclicity of Mgr on 2- and 3-strand torus braids; the load-bearing 4-strand acyclicity is proven in-paper (Proposition 3.6), so that citation is not load-bearing. Flagged explicitly per the manuscript-review rule: Lemma 5.1's proof omits its final case — 'We omit the details of the last case: showing that Equivalence 24 holds for a in (g4 o IB o g2 o IA)({e})' — and the fallback verification (braidalgo.py) only covers n <= 50. Because Lemma 5.2, Lemma 5.3, Lemma 4.5, Lemma 6.2, and hence both recursions of Theorem 1.1 and the induction in Theorem 1.2 for n >= 83, depend on the complete equality W = U, this omission is a genuine correctness/completeness risk for the central claims; however it is not circular, since W is defined independently of U (Equation 22) and no statement reduces to its own input. That risk belongs in a correctness assessment, not in the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rely on no fitted numeric constants. The mathematical inputs are standard Khovanov/Bar-Natan theory, algebraic discrete Morse theory, the unproved classification of unmatched cells, and the correctness of external base-case computations.

assumptions (3)
  • domain assumption The omitted case of Lemma 5.1, classifying unmatched cells for a ∈ (g4 ◦ IB ◦ g2 ◦ IA)({e}), holds as stated.
    The proof says 'We omit the details of the last case' and only gives code verification for n ≤ 50; all recursive constructions use W = U.
  • ad hoc to paper Lemma 5.2: the longest W-word containing no A^2, B^2 or C^2 subword has length 3·23.
    This bound is asserted 'by staring at the definition of W' and underlies Proposition 4.1, so it is load-bearing and not proven in the text.
  • domain assumption The computer base cases for n = 28,...,82 computed by Khoca are correct.
    Theorem 1.2 uses these bases as the starting point for induction; an error in any base case would propagate.

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Pith. "Pith review of Morse matchings and Khovanov homology of 4-strand torus links." pith.science (2026). https://pith.science/paper/ASRFQETE

@misc{pith2026250715060,
  author       = {Pith},
  title        = {Pith review of: Morse matchings and Khovanov homology of 4-strand torus links},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASRFQETE}},
  note         = {Machine review of arXiv:2507.15060}
}
abstract

Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty)$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $\lambda$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$.

Figures

Figures reproduced from arXiv: 2507.15060 by the authors.

Figure 1
Figure 1. For n ≥ 14, the unreduced integer Khovanov homology Khi,j (T(4, −2n)) (top left) and Khi,j (T(4, −2n−1)) (top right) and the reoccurring block (bottom). The homology remains unknown in the gray area above the purple blocks and it vanishes below them. More precisely, the top left table describes Khi,j (T(4, −2n)) only when i ∈ [−4n + 11, −42] and j ≤ 3 2 i − 6n + 5. The top right table describes Khi,j (T(4, −2n − 1))… view at source ↗
Figure 2
Figure 2. The dotted relations of Mat(Cob3 • (2n)). where Lc is the set of loops in c and c ′ is the planar diagram of c with all of the loops removed. We denote Ψc for the codomain of Ψc and graphically represent a direct summand of Ψc by coloring the loops contained in K with red and those in Lc \ K blue. By acting diagonally, Ψ can be extended to formal direct sums and for any object a. ⃝ ∅{−1} ∅{1} ⃝ ⊕ ⃝ ⃝ ⃝ ⃝ ⊕ [PITH_FU… view at source ↗
Figure 3
Figure 3. For a diagram which consists of only a single circle: delooping isomorphism Ψ and its [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Morse presentation of an oriented tangle with 5 Morse layers. On layer 2 we see a positive [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: A local dictionary for matrix elements of Ψ [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Using green as a placeholder color for red, blue or black, we can divide the isomorphisms [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: A morse presentation of a tangle T (left) and its delooped hypercube complex ΨJTK (right). The complex has two isomorphisms, f : 00• → 10• and g : 10• → 11• which satisfy i(f) = 2, i(g) = 1 and u(f) = u(g) = 3. Both isomorphism get matched in Mgr and hence MgrΨJTK cons…
Figure 8
Figure 8. Figure 8: The three periodic building blocks A, B and C of unmatched cells of G(ΨJ(σ1σ2σ3) nK, Mgr). The maps an, bn and cn from Proposition 4.1 take a diagram w of Un which contains at least one of the periodic building blocks A, B and C and duplicates the first instance of thi…
Figure 9
Figure 9. Figure 9: Our convention for composing 8-ended tangles (left) and an example link ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: On the left, the pair (D2 ×[0, 1], {(− 1 2 , 0)∪( 1 2 , 0)}×[0, 1]) which is our model for rational tangles. In the center and on the right we see two rational tangles. Switching one with the other in some region of a knot is a proper rational tangle replacement, sinc…
Figure 11
Figure 11. Figure 11: Two global environments r, s ∈ R. Since r ∼ s we can apply colr to s, and the stronger relation applies: s ≈ colr(s). the codomain G(ΨJ(σ1σ2σ3) mK, Mgr). The orientation of a → b might change under Λs,t,m, as a pri￾ori it cannot be guaranteed that both or neither of t…
Figure 12
Figure 12. Figure 12: Local moves part I. The typographic naming of the moves tries to mimic their patterns [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]
Figure 13
Figure 13. Figure 13: Local moves part II. The moves ▽, ▽ −1 , ( are always located at the very top of the braid which is drawn with an orange line in the pictures. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_13.png]
Figure 14
Figure 14. Figure 14: 37 [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]
Figure 14
Figure 14. Figure 14: Classification of reversed arrow with u evaluated in the repeating region. The first Morse layer of the pictured subword z k = (001010) k is 3m − 5 or 3m − 8 in all of the diagrams. Notice that all of the arrow types of [PITH_FULL_IMAGE:figures/full_fig_p038_14.png]
Figure 15
Figure 15. Figure 15: The braid diagram Γ and the unique cycle [PITH_FULL_IMAGE:figures/full_fig_p049_15.png]
Figure 16
Figure 16. Figure 16: For n ≥ 28 the unreduced integral Khovanov homology Khi,j (T(4, −n)) in the lowest non-trivial homological degrees. Outside the marked entries the homology vanishes for i ≤ −8n+10, i ≤ −8n + 6 and i ≤ −4n + 6 respectively. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_16.png]
Figure 17
Figure 17. Figure 17: For n ≥ 28 the unreduced integral Khovanov homology Khi,j (T(4, −n)) in the highest non-trivial homological degrees. Outside the marked entries the homology vanishes for i ≥ −41. 51 [PITH_FULL_IMAGE:figures/full_fig_p051_17.png]

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