A note on the X-torsion order of a knot

Dirk Schuetz

A note on the X-torsion order of a knot

T0 review · 0 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a knot $K$ and a field $F$ of characteristic different from $2$, the $X$-torsion order $\mathrm{xo}_F(K)$ equals one less than the number of pages of the reduced Bar-Natan–Lee–Turner spectral sequence.

desk verdict Short, solid paper: Theorem 1.1 linking X-torsion order to the reduced Bar-Natan–Lee–Turner page count is new and useful, and the few terse proof steps are not load-bearing. read the letter →

arxiv 2412.05156 v1 pith:O74TTHVG submitted 2024-12-06 math.GT

Dirk Schuetz This is my paper · ORCID

classification math.GT MSC 57K1857K10

keywords X-torsionorderKhovanovhomologyLeeBar-Natanspectralsequencetorsioninvariantknot

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

The paper establishes that the $X$-torsion order of a knot—a torsion invariant defined through a generalized Lee complex over a field—can be computed as one less than the number of pages of the reduced Bar-Natan–Lee–Turner spectral sequence. This turns a module-theoretic quantity into a page count that existing software can compute. The paper uses this equivalence to calculate $X$-torsion orders for knots up to 16 crossings, finding examples of order 4 and new field dependence. If the theorem is correct, a broad class of torsion-order computations becomes routine.

What carries the argument

The central object is the ring isomorphism $\Phi: F[X,H]/(X^2-XH) \to F[X,T^{1/2}]/(X^2-T)$ defined by $H \mapsto T^{1/2}$ and $X \mapsto (X+T^{1/2})/2$. It relates the Bar-Natan Frobenius system to a half-twisted Lee system and, after the twist by $1/2$, induces a grading-preserving isomorphism of chain complexes. The isomorphism transfers the Smith normal form decomposition of the Lee complex—where torsion appears as powers of $X$—into complexes over $F[H]$ whose differentials are powers of $2X-H$; in the reduced complex these become powers of $H$. The largest such power is exactly the number of pages before the reduced Bar-Natan–Lee–Turner spectral sequence collapses, which is the equality stated in the theorem.

What would settle it

Compute the $X$-torsion order directly from the Lee complex over $F[X]$ and separately compute the page count of the reduced Bar-Natan–Lee–Turner spectral sequence for a small knot such as the trefoil or $T(5,6)$; any mismatch would disprove the theorem, and a chain-level comparison of the two complexes for a single nontrivial diagram would test the isomorphism itself.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every knot $K$ and every field $F$ of characteristic different from $2$, $\mathrm{xo}_F(K) = \widetilde{\mathrm{pg}}_F(K) - 1$. Here $\mathrm{xo}_F(K)$ is the smallest $n$ such that $X^n$ annihilates the torsion submodule of Lee homology over $F[X]$, and $\widetilde{\mathrm{pg}}_F(K)$ is the number of pages of the reduced Bar-Natan–Lee–Turner spectral sequence. The proof passes through a ring isomorphism that identifies the Bar-Natan complex with the Lee complex after a half-twist, then matches the Smith normal form decomposition of the Lee complex with the page count of the reduced spectral sequence. Computations reported in the paper include knots with $X$-torsion order $4$ and cases where the value changes with the field, such as $T(8,9)$ having different orders over $\mathbb{Q}$ and $F_7$ despite equal Khovanov Betti numbers.

Load-bearing premise

The load-bearing premise is that the ring isomorphism $\Phi$ preserves the gradings and Frobenius structure well enough to identify, after a twist by $1/2$, the Bar-Natan chain complex with the Lee chain complex; if that graded identification fails, the equality between the $X$-torsion order and the page count does not follow.

Editorial extensions

If this is right

The $X$-torsion order becomes available from existing computations of the reduced Bar-Natan–Lee–Turner spectral sequence, bypassing direct Lee-complex decompositions.
The equality extends the invariant to $F_2$, where it agrees with the $H$-torsion order.
The computed examples show that the $X$-torsion order is not determined by Khovanov Betti numbers: $T(8,9)$ has the same Betti numbers over $\mathbb{Q}$ and $F_7$ but different $X$-torsion orders.
The counterexample knot to the knight-move conjecture over $\mathbb{Q}$ has $X$-torsion order $2$ over $F_2$ and $F_3$, so it does not violate the conjecture in those characteristics.
A variant of that construction gives a knot with $X$-torsion order $4$, the largest value found in the paper's range.

Reading between the lines

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

The same isomorphism could be used to show that other torsion orders arising from Frobenius deformations are all captured by one page-count formula, possibly explaining patterns in the known torsion invariants.
If the grading-preserving isomorphism extends to links with basepoints, the basepoint-dependent $X$-torsion order for links might also be a reduced spectral sequence page count; the split-link-with-unknot example in the paper would be a natural test case.
Because the page count is bounded by the homological width of the diagram, the identification may yield new upper bounds on $X$-torsion order in terms of crossing number or braid index.
A direct chain-level check of $\Phi$ on a small knot such as the trefoil would independently test the most delicate step of the proof, since both sides of the isomorphism are explicitly computable.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

This is a short, honest paper. The main result, Theorem 1.1, identifies the X-torsion order xo_F(K) with ~pg_F(K) − 1, the page count of the reduced Bar-Natan–Lee–Turner spectral sequence minus one. That is a new equivalence, and it is genuinely useful: the page count is already implemented in knotjob, so computing xo_F becomes cheap. The paper also turns up two concrete phenomena worth knowing about: the first knot with X-torsion order 4, and T(8,9) where xo_Q = 2 < 3 = xo_F7 even though the Khovanov Betti numbers are the same over both fields. The characteristic dependence of torsion is exactly the kind of thing the field needs more examples of.

What the paper does well: the proof is short and mostly self-contained. The Smith normal form decomposition (1) is standard, and the ring isomorphism Φ is written down explicitly. The paper is careful to note that Φ is not literally a Frobenius isomorphism, only one after twisting by 1/2. That step is the most compressed part — it is cited to [Kho06] rather than shown. But the stress-test note is right that this does not matter: the twist multiplies the differential entries by the unit 1/2, and unit rescaling preserves all determinantal ideals, hence the invariant factors k_i. The paper could have said that in one line, and its absence is an expository gap, not a mathematical one.

Soft spots, in proportion: the computational section is a census without a data table. The paper names the five 15-crossing knots and the 111 16-crossing knots but does not list them; a reader who wants to verify or reuse the data has to rerun knotjob. That is fine for a note, but a referee should ask for the lists or a link to the data. The extension of xo to F2 by definition (Deﬁnition 4.1) is a stipulation rather than a theorem; the paper is upfront about it, and the agreement with Alishahi's H-torsion order is stated without proof. Again, acceptable in a note, but worth flagging. Also, the claim that the spectral sequence collapses after k steps is justified in a sentence; that is consistent with the decomposition, but a referee may want the page-count statement formalized.

The citation pattern looks reasonable: the key references (Kho06, Lee05, LS22, Tur20) are the right ones, and the new claims are not hidden. No fitting, no circularity; the computations follow from the theorem.

Bottom line: this is a useful, correct note. I would send it to a serious referee. Recommended: accept after minor revisions, mainly asking for data lists and the one-line unit-invariance clarification.

Referee Report

0 major / 4 minor

Summary. The paper proves that for a knot K and a field F of characteristic different from 2, the X-torsion order xo_F(K) equals ~pg_F(K) − 1, where ~pg_F(K) is the number of pages of the reduced Bar-Natan–Lee–Turner spectral sequence. The proof uses an explicit ring isomorphism between the Bar-Natan and Lee Frobenius systems, the Smith normal form decomposition of the Lee complex, and a comparison of the reduced complexes. The paper also computes xo_F for knots up to 16 crossings, finding examples of X-torsion order 4 and field dependence, and extends the definition to F2.

Significance. If correct, Theorem 1.1 provides a direct and computationally accessible way to compute the X-torsion order via a spectral sequence. The calculations are a strength: they reveal field dependence in surprising places (e.g., the Manolescu–Marengon knot with xo_Q=3 but xo_F2=xo_F3=2, and the torus knot T(8,9) with xo_Q=2 < xo_F7=3 despite equal Betti numbers). The proof is elegant and mostly self-contained. The one step that needs attention is the 'twist by 1/2' in Section 3; it is mathematically sound but terse, and a short clarification would remove the main ambiguity.

minor comments (4)

[Section 3] The isomorphism Φ is said to induce a grading-preserving isomorphism from CBN(D;F[H]) to CLee(D;F[T^{1/2}]) only after twisting the Lee system by 1/2, but the twist is not defined and the proof subsequently applies the Smith normal form decomposition (1) of the untwisted Lee complex. Please specify the twisted Frobenius system and add a sentence noting that multiplying the comultiplication by the unit 1/2 leaves the invariant factors k_i in (1) unchanged (units preserve the determinantal ideals of the differential matrices), so the decomposition applies to the twisted complex as well.
[Section 3] The assertion that the described algebraic decomposition implies that the reduced Bar-Natan–Lee–Turner spectral sequence collapses after k steps is stated without proof; a brief argument or a precise reference would make Theorem 1.1 self-contained.
[Section 4] The sentence 'There are also 111 knots with 16 crossings such that xo_Q(K) = xo_F2(K)' is ambiguous and likely contains a typo: if 111 is the number of knots where equality holds, the total number of 16-crossing knots should be stated, and if 111 is the number of exceptions, the inequality should be written. Please clarify.
[Section 4] Please clarify whether the Manolescu–Marengon knot K and its variation K′ lie within the set of knots with up to 16 crossings, since the text first says that no X-torsion order exceeds 2 for knots up to 16 crossings and then gives xo_Q(K)=3 and xo_Q(K′)=4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 derives the equality from an explicit chain complex isomorphism and an independent Smith normal form decomposition; no prediction reduces to a fitted input or self-citation.

full rationale

The paper's central claim, xo_F(K) = ~pg_F(K) - 1, is derived by comparing two explicit chain complexes, not by defining one invariant in terms of the other. The Lee complex is decomposed via the Smith normal form statement in equation (1), which is an independent structural fact about complexes over the Euclidean domain F[X]. The Bar-Natan complex is then related to the (half-twisted) Lee complex through the explicit ring isomorphism Phi in Section 3, with the key identities Phi(2X-H) = X and Phi(X-H) = (1/2)(X - T^(1/2)) written out. The 'twist by 1/2' mentioned there is a rescaling by a unit, and the paper's own comparison shows the differentials correspond after this unit rescaling; unit multiplication preserves the invariant factors, so this does not smuggle the conclusion into the premise. The reduction to the reduced complex and the identification of the spectral sequence page count with the maximum k_i are transparent algebraic steps, not fitted parameters or renamed inputs. The only self-referential-looking passage, Definition 4.1, extends xo to F2 by setting it equal to ~pg_F2(K) - 1; that is an explicit definitional extension, explicitly flagged as such, not a disguised derivation of Theorem 1.1. Citations to Khovanov, Lee, Turner, and Lipshitz-Sarkar are used for standard background facts and do not carry the load of Theorem 1.1. Since the theorem is self-contained against the internal decomposition (1) and the explicit isomorphism, the appropriate finding is no circularity.

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

The central claim rests on three unproved background facts: the Smith Normalization decomposition over F[X], the chain complex isomorphism induced by Φ with a twist, and the correspondence between the model complex and the spectral sequence page count. None of these are fitted to the examples; they are standard tools from the literature, but the latter two are only cited, not proved in the paper.

assumptions (3)

standard math F[X] is a PID, so every finite-rank F[X]-chain complex decomposes via Smith Normalization into free and two-term torsion summands.
Used in Section 3 to obtain the decomposition (1) of C_Lee(D;F[T]) into F[X] summands with differential X^{k_i}.
domain assumption The ring isomorphism Φ sends the Bar-Natan Frobenius algebra to a twist of the Lee Frobenius algebra over F[T^{1/2}], and this induces a grading-preserving isomorphism of Khovanov chain complexes.
Invoked in Section 3 as 'compare [Kho06]' to identify C_BN(D;F[H]) with C_Lee(D;F[T^{1/2}]). This is the key algebraic bridge.
domain assumption The reduced Bar-Natan-Lee-Turner spectral sequence is the H-adic filtration spectral sequence, and the model complex F[H] --H^k--> F[H] makes the sequence stabilize after exactly k+1 pages.
Used in Section 3, final paragraph, to translate the model decomposition into ~pg_F(K) = max k_i + 1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A note on the X-torsion order of a knot." pith.science (2026). https://pith.science/paper/O74TTHVG

@misc{pith2026241205156,
  author       = {Pith},
  title        = {Pith review of: A note on the X-torsion order of a knot},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O74TTHVG}},
  note         = {Machine review of arXiv:2412.05156}
}

abstract

We show that the $X$-torsion order of a knot, which is defined in terms of a generalised Lee complex, can be calculated using the reduced Bar-Natan--Lee--Turner spectral sequence. We use this for extensive calculations, including an example of $X$-torsion order $4$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 10 canonical work pages

[1]

254 (2019), 29--38

Akram Alishahi and Nathan Dowlin, The L ee spectral sequence, unknotting number, and the knight move conjecture , Topology Appl. 254 (2019), 29--38. 3894208

work page 2019
[2]

Akram Alishahi, Unknotting number and K hovanov homology , Pacific J. Math. 301 (2019), no. 1, 15--29. 4007369

work page 2019
[3]

Knot Theory Ramifications 16 (2007), no

Dror Bar-Natan, Fast K hovanov homology computations , J. Knot Theory Ramifications 16 (2007), no. 3, 243--255. 2320156

work page 2007
[4]

Lowrance, Radmila Sazdanovi\'c, and Melissa Zhang, On K hovanov homology and related invariants , Research directions in symplectic and contact geometry and topology, Assoc

Carmen Caprau, Nicolle Gonz\'alez, Christine Ruey Shan Lee, Adam M. Lowrance, Radmila Sazdanovi\'c, and Melissa Zhang, On K hovanov homology and related invariants , Research directions in symplectic and contact geometry and topology, Assoc. Women Math. Ser., vol. 27, Springer, Cham, [2021] 2021, pp. 273--292. 4417719

work page 2021
[5]

Onkar Singh Gujral, Ribbon distance bounds from bar-natan homology and -homology, 2020

work page 2020
[6]

Damian Iltgen, Lukas Lewark, and Laura Marino, Khovanov homology and rational unknotting, ArXiv e-print 2110.15107 (2021)

arXiv 2021
[7]

Mikhail Khovanov, Link homology and F robenius extensions , Fund. Math. 190 (2006), 179--190. 2232858

work page 2006
[8]

Eun Soo Lee, An endomorphism of the K hovanov invariant , Adv. Math. 197 (2005), no. 2, 554--586. 2173845

2005

Show all 14 references

[9]

Lukas Lewark, Laura Marino, and Claudius Zibrowius, Khovanov homology and refined bounds for gordian distances, ArXiv e-print 2409.05743 (2024)

2024 arXiv
[10]

Robert Lipshitz and Sucharit Sarkar, A mixed invariant of nonorientable surfaces in equivariant K hovanov homology , Trans. Amer. Math. Soc. 375 (2022), no. 12, 8807--8849. 4504654

2022
[11]

Ciprian Manolescu and Marco Marengon, The knight move conjecture is false, Proc. Amer. Math. Soc. 148 (2020), no. 1, 435--439. 4042864

2020
[12]

Sucharit Sarkar, Ribbon distance and K hovanov homology , Algebr. Geom. Topol. 20 (2020), no. 2, 1041--1058. 4092319

2020
[13]

Knot Theory Ramifications 29 (2020), no

Paul Turner, Khovanov homology and diagonalizable F robenius algebras , J. Knot Theory Ramifications 29 (2020), no. 1, 1950095, 10. 4079621

2020
[14]

Knot Theory Ramifications 31 (2022), no

Zipei Zhuang, Knot cobordism and L ee's perturbation of K hovanov homology , J. Knot Theory Ramifications 31 (2022), no. 2, Paper No. 2250012, 6. 4420588

2022

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

254 (2019), 29--38

Akram Alishahi and Nathan Dowlin, The L ee spectral sequence, unknotting number, and the knight move conjecture , Topology Appl. 254 (2019), 29--38. 3894208

work page 2019

[2] [2]

Akram Alishahi, Unknotting number and K hovanov homology , Pacific J. Math. 301 (2019), no. 1, 15--29. 4007369

work page 2019

[3] [3]

Knot Theory Ramifications 16 (2007), no

Dror Bar-Natan, Fast K hovanov homology computations , J. Knot Theory Ramifications 16 (2007), no. 3, 243--255. 2320156

work page 2007

[4] [4]

Lowrance, Radmila Sazdanovi\'c, and Melissa Zhang, On K hovanov homology and related invariants , Research directions in symplectic and contact geometry and topology, Assoc

Carmen Caprau, Nicolle Gonz\'alez, Christine Ruey Shan Lee, Adam M. Lowrance, Radmila Sazdanovi\'c, and Melissa Zhang, On K hovanov homology and related invariants , Research directions in symplectic and contact geometry and topology, Assoc. Women Math. Ser., vol. 27, Springer, Cham, [2021] 2021, pp. 273--292. 4417719

work page 2021

[5] [5]

Onkar Singh Gujral, Ribbon distance bounds from bar-natan homology and -homology, 2020

work page 2020

[6] [6]

Damian Iltgen, Lukas Lewark, and Laura Marino, Khovanov homology and rational unknotting, ArXiv e-print 2110.15107 (2021)

arXiv 2021

[7] [7]

Mikhail Khovanov, Link homology and F robenius extensions , Fund. Math. 190 (2006), 179--190. 2232858

work page 2006

[8] [8]

Eun Soo Lee, An endomorphism of the K hovanov invariant , Adv. Math. 197 (2005), no. 2, 554--586. 2173845

2005