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The CN matrix of a pure braid projection

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For six strands, a matrix is the crossing-number matrix of a pure braid projection exactly when it is even, symmetric, and T0.

desk verdict Solid n=6 extension of the CN/OU/crossing matrix characterization, let down only by an unreproducible computer check in the key exhaustive step. read the letter →

arxiv 2506.08659 v3 pith:FHEUUOZW submitted 2025-06-10 math.GT

Yuko Ozawa , Ayaka Shimizu , Yoshiro Yaguchi This is my paper
classification math.GT MSC 57K1020F36
keywords CNmatrixpurebraidprojectiondiagramcrossingOUT0BW-ladderpositive
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 proves that, for six strands, three conjectures about braid matrices are true. Its central object is the CN matrix of a braid projection, whose $(i,j)$ entry is the number of crossings between the $i$-th and $j$-th strands. The main result says a six-by-six non-negative integer matrix is the CN matrix of some pure six-braid projection exactly when it is even, symmetric, and satisfies the $T_0$ condition: for $i

What carries the argument

The BW-ladder diagram carries the argument. For a $(0,2)$-matrix $M$, one draws a black edge between strands $i$ and $j$ whenever the entry is 2; the ladder moves $L_1$ through $L_9$ are local rewrites that move black and white edges past each other. If a B-ladder diagram can be transformed into a W-ladder diagram, then $M$ is CN-realizable, because the white edges are the crossings of a braid projection. Section 4 catalogues formations—snake, hang-glider, and loupe—each with explicit ladder-move proofs of CN-realizability, and Section 5 packages them into the 128 configurations used in the exhaustive 6x6 check. The $T_0$ condition is the necessary side: it forbids the pattern of two zero entries forcing a non-zero entry.

What would settle it

Re-run the elimination of the 4,824 $T_0$ $(0,2)$-matrices: if any one of them lacks all of the configurations a1-a4, b1-b3, c1-c100, and d1-d21 and their reverses, and its B-ladder diagram cannot be transformed into a W-ladder diagram by the ladder moves, then the characterization of CN-realizable 6x6 matrices is false. A single 6x6 even symmetric $T_0$ matrix whose B-ladder diagram provably admits no ladder-move reduction would also disprove Proposition 25.

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Extended reading notes

Core claim

Proposition 24 is the central claim: a 6x6 upper triangular $(0,2)$-matrix—one with entries only 0 or 2 above the diagonal—is CN-realizable if and only if it is $T_0$. Proposition 25 then states the consequence on the paper's own terms: a non-negative integer 6x6 matrix $M$ is the CN matrix of some pure 6-braid projection if and only if $M$ is even, symmetric, and $T_0$. From this, Corollary 4 characterizes the OU matrix of a pure 6-braid diagram by the condition that $M+M^{T}$ is even and $T_0$, and Corollary 5 characterizes the crossing matrix of a positive pure 6-braid as a non-negative integer $T_0$ symmetric matrix. Together these prove Theorem 1: Conjectures 1, 2, and 3 are true for $n \le 6$.

Load-bearing premise

The load-bearing step is the computer check that every one of the 4,824 six-by-six matrices with entries 0 or 2 that satisfy the $T_0$ condition contains at least one of the listed realizable configurations or its reverse; if that exhaustive check missed a case, Proposition 24 and Theorem 1 would fail.

Editorial extensions

If this is right

  • Every 6x6 even symmetric $T_0$ matrix is realized by some pure 6-braid projection, so for six strands the geometric existence question has a purely algebraic answer.
  • A 6x6 non-negative integer matrix $M$ is the OU matrix of a pure 6-braid diagram exactly when $M+M^{T}$ is even and $T_0$.
  • A 6x6 integer matrix is the crossing matrix of a positive pure 6-braid exactly when it is non-negative, $T_0$, and symmetric.
  • Because the same characterization was already known for $n \le 5$, the result gives a uniform statement for all braid projections with at most six strands.

Reading between the lines

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

  • As an extension beyond the paper's own claims, the same finite configuration-check could be attempted for $n=7$: the paper counts 96,428 $T_0$ $(0,2)$-matrices of size 7, so a larger catalogue might settle the conjecture before a general proof is available.
  • If Conjecture 4, the T-structure condition, is true, the snake, hang-glider, and loupe formations would all be instances of one graph-theoretic sufficient condition, potentially opening an inductive proof for all $n$.
  • The paper proves that the crossing-matrix conjecture follows from the CN conjecture at any fixed $n$; a future proof of Conjecture 3 for arbitrary $n$ would therefore settle all three conjectures at once.
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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

2 major / 3 minor

Summary. The paper studies CN matrices of pure braid projections, the symmetric integer matrices whose (i,j) entry counts crossings between the i-th and j-th strands. It introduces several explicit formation types (snake, hang-glider, loupe) and proves via BW-ladder moves that matrices with these formations are CN-realizable. It then defines a large family of CN-realizable configurations for 6x6 T0 (0,2)-matrices and states the main theorem: for n <= 6, the CN matrix conjecture, the OU matrix conjecture, and the positive pure braid crossing matrix conjecture all hold. The proof of the n=6 case rests on Proposition 24, whose sufficiency direction is justified by the assertion that a computer check confirmed that all 4,824 T0 (0,2)-matrices contain at least one of the listed configurations or their reverses.

Significance. If the result is fully supported, the paper settles the three conjectures for n <= 6, extending the previously known n <= 5 cases. The individual formation proofs via ladder moves are explicit and appear checkable, and the reductions in Section 6.2 and 6.3 from a CN-matrix characterization to the OU and crossing matrix characterizations are clean. However, the decisive exhaustive step is not independently verifiable: the paper does not provide the computer program, the data, or a machine-checkable certificate, and the configuration list is given only as human-readable figures. The significance of the paper is therefore conditional on filling this reproducibility gap.

major comments (2)
  1. [§6.1, Proposition 24] The proof of Proposition 24 relies entirely on the assertion that all 4,824 6x6 upper triangular T0 (0,2)-matrices contain at least one of the listed CN-realizable configurations or their reverses, and that this was "confirmed by eliminating the matrices ... by computer". No code, data, or certificate is provided, and the count 4,824 alone does not establish coverage. If even one T0 matrix were missed by the configuration list, Proposition 24 would be false, and with it Proposition 25, Corollaries 4 and 5, and Theorem 1 would fall. This is a load-bearing exhaustive claim that cannot be replayed from the manuscript as written; a reproducible computer program, the elimination output, or an explicit certificate (for example, a list assigning to each matrix a witnessing configuration) is needed.
  2. [§5, Propositions 22 and 23] The configurations c1-c100 and d1-d21 are presented only as figures, with no formal specification in a machine-readable form. Since the computer elimination in Proposition 24 must test whether a matrix "has at least one of the CN-realizable configurations", the absence of a precise specification makes the exhaustive step even harder to verify. A machine-readable list of the configurations, or at least a precise coordinate description, should accompany the paper so that the claimed elimination can be independently checked.
minor comments (3)
  1. [§2, Lemma 1] The statement of Lemma 1 says M(i,j) = 0 when i < I or j > I - m - 1, but Proposition 9 and the surrounding text indicate the intended condition is j > I + m - 1. As printed, the lemma is trivial for many parameter choices; please correct this apparent typo.
  2. [Throughout] There are several typographical errors that should be fixed: "edegs" in the proof of Proposition 14, "confugurations" in the caption of Figure 20, "exsist" in the footnote to Conjecture 4, "fromation" in the captions of Figures 25 and 26, and "calcurated" in Remark 1.
  3. [§4.4, Conjecture 4] The assertion that "all the matrices of a snake, hang-glider or loupe formation have a T-structure" is stated without proof. It would be helpful to add a brief justification or to mark it explicitly as an observation, since Section 4.4 is otherwise somewhat detached from the main theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the n=6 characterization is built from explicit ladder-move realizations and a brute-force coverage check, with no fitted parameter or definitional equivalence forcing the conclusion.

full rationale

The paper's central claim, Proposition 24, is that every 6x6 upper triangular T0 (0,2)-matrix is CN-realizable. The necessity direction is Proposition 5 from [6]; the sufficiency direction is a finite exhaustive computer check: 'All the 6 x 6 upper triangular T0 (0,2)-matrices, whose number is 4,824, have at least one of the CN-realizable configurations or their reverses of Examples 5, 6 or Propositions 22, 23. It was confirmed by eliminating the matrices that have the CN-realizable configurations from the list of the 4,824 T0 matrices by computer.' The individual configurations are proven CN-realizable by explicit ladder-move arguments (Propositions 13-20 and Examples 5-6), not by assuming the target characterization. The reduction from the (0,2) case to arbitrary even matrices uses Proposition 7 of [6], and the reductions from Conjecture 3 to Conjectures 2 and 1 in Propositions 26 and 27 are formal implications with proofs supplied here. Earlier results from [6] are used as lemmas, including Theorem 2 for n<=5; this is a legitimate prior theorem, not an import of the n=6 conclusion. The main weakness is that the exhaustive coverage assertion in Proposition 24 is not accompanied by code or data, and the footnote notes that the previous arXiv version used a different configuration set, so the enumeration cannot be replayed from the paper alone. That is a reproducibility and verification gap, not circularity: no matrix class is defined in terms of the target result, no fitted parameter is relabeled as a prediction, and no uniqueness claim is imported from the authors' prior work to force the choice.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof rests on a small number of background facts from [6] plus one unverifiable computational assertion (the exhaustive check of all 4,824 T0 matrices). No free parameters or new entities are introduced.

assumptions (2)
  • ad hoc to paper Computer verification result: Every 6x6 upper triangular T0 (0,2)-matrix has at least one of the CN-realizable configurations a1-a4, b1-b3, c1-c100, d1-d21 (or their reverses).
    This is the load-bearing finite check stated in the proof of Proposition 24 and credited to 'computer' but never documented with code or data, so the reader must take it on faith.
  • domain assumption Results from the authors' earlier work [6] are valid, including Propositions 5, 7, 8, 10, 11, 12 and Corollary 1 on CN matrices, ladder moves, and the OU matrix.
    The paper's arguments rely repeatedly on these previously published theorems by the same authors without reproving them; they are not derived in this preprint.

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Pith. "Pith review of The CN matrix of a pure braid projection." pith.science (2026). https://pith.science/paper/FHEUUOZW

@misc{pith2026250608659,
  author       = {Pith},
  title        = {Pith review of: The CN matrix of a pure braid projection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FHEUUOZW}},
  note         = {Machine review of arXiv:2506.08659}
}
abstract

The CN matrix of an $n$-braid projection $B$ is an $n \times n$ matrix such that each $(i,j)$ entry indicates the number of crossings between $i^{th}$ and $j^{th}$ strands of $B$. In this paper, several patterns of an $n \times n$ matrix to be a CN matrix are discussed, and the CN matrix of a pure 6-braid projection is characterized. As an application, the OU matrix of a pure 6-braid diagram and the crossing matrix of a positive pure 6-braid are also characterized.

Figures

Figures reproduced from arXiv: 2506.08659 by the authors.

Figure 1
Figure 1. The crossing matrix C ( b) and the OU matrix U ( b) of a 4-braid diagram b and the CN matrix N ( B) of a 4-braid projection B . crossing of a braid diagram b has the sign as indicated in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. A positive crossing on the left and a negative crossing on the right. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. An example of Proposition 9 with m = 3, n = 6, I = 3. For 6 × 6 matrices, we have the following lemma by Proposition 9 and Theorem 2. Lemma 1. Let M be a 6 × 6 T0 strictly upper triangular (0, 2)-matrix such that M(i, j) = 0 when i < I or j > I − m − 1 for some integers 1 ≤ m ≤ 5 and 1 ≤ I ≤ 7 − m. Then M is CN-realizable. 3 BW-ladder diagram In this section, we review the BW-ladder diagram which was introduced in [… view at source ↗
Figures from the paper (25 more)
Figure 4
Figure 4. Figure 4: Ladder moves. Proposition 10 ([6]). A strictly upper triangular (0, 2)-matrix M is CN￾realizable if a B-ladder diagram of M can be transformed into a W-ladder diagram by a finite sequence of ladder moves. Example 1. The (0, 2)-matrix M in [PITH_FULL_IMAGE:figures/full…
Figure 5
Figure 5. Figure 5: The matrix M is CN-realizable since the B-ladder diagram of M is transformed into a W-ladder diagram. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: The r(2, 4)-, c(3, 6)- and an rc(2, 6)-formations. Proposition 13. Any strictly upper triangular matrix of r- or c-formation is a CN-realizable matrix. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: A matrix of an rc(1, 6)-formation is CN-realizable. Definition 11. Let (α1, α2) = (c, r), (c, rc), (rc, r) or (rc, rc). An n×n strictly upper triangular matrix M is said to have an α1(k, l)-α2(l−1, m)-formation (or simply α1-α2-formation) if M = M1 + M2 − M3, where M1 …
Figure 8
Figure 8. Figure 8: c-r-, c-rc-, rc-r- and rc-rc-formations. Proposition 15. Any strictly upper triangular matrix of an α1-α2-formation is a CN-realizable matrix, where (α1, α2) = (c, r), (c, rc), (rc, r) or (rc, rc). Proof. Let M be a matrix of the c(k, l)-rc(l − 1, m)-formation. Take a …
Figure 9
Figure 9. Figure 9: A matrix of a c(1, 4)-rc(3, 7)-formation is a CN-realizable matrix. Proposition 16. Let M1 (resp. M2) be an n×n matrix such that M1(l−2, l+ 1) = 2 (resp. M2(l − 2, l + 2) = 2) and the other entries are zero. (1) Let M be an n×n matrix of a c(k, l)-r(l −1, m)-formation …
Figure 10
Figure 10. Figure 10: A matrix of a c(1, 4)#r(3, 7)-formation is a CN-realizable matrix. Ignore the orange-colored broken edges if M(2, 6) = 0. Proposition 17. Any strictly upper triangular matrix of an H-formation is a CN-realizable matrix. Proof. Let M be an n × n matrix of an H(k, l; m)…
Figure 11
Figure 11. Figure 11: A matrix of an H(1, 7; 3)-formation is a CN-realizable matrix. Ignore the broken edges if the corresponding entries are 0 in the matrix. Proof. For the B-ladder diagram in the proof of Proposition 17, add B k−1 k+1 (resp. B l−1 l+1 ) to the top (resp. to the bottom), …
Figure 12
Figure 12. Figure 12: A matrix of an L1(2, 7)-formation is a CN-realizable matrix. S 2 7 , S 3 7 , S 4 7 are highlighted in yellow. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: A matrix of an L2(2, 7)-formation is a CN-realizable matrix. S 2 7 and S 3 7 are highlighted in yellow. Proof. Let S k l = Bk k+1Bk k+4Bk k+5 . . . Bk l B k+2 k+4B k+2 k+5 . . . Bk+2 l B k+3 l B k+3 l−1 . . . Bk+3 k+4 . Then S k l is transformed into Wk k+1Wk+3 k+4 Wk…
Figure 14
Figure 14. Figure 14: A matrix of an L3(2, 8)-formation is a CN-realizable matrix. S 2 8 , S 3 8 , S 4 8 are highlighted in yellow. Definition 17. For a grid alignment G(M), a graph on G(M) is a graph in the grid that is obtained from G(M) by connecting some vertices by edges with the foll…
Figure 15
Figure 15. Figure 15: (1): A (0, 2)-matrix M. (2): The grid alignment G(M) of M. (3): A graph g on G(M). j = j0 > j1 > j2 > · · · > jn, where each pair of vertices V (i, jl−1) and V (i, jl) share an edge. We call the sequence the horizontal path for V (i, j) with length n. In the same way,…
Figure 16
Figure 16. Figure 16: On (C2) and (C3) [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: A graph which fails to have a T-structure. [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: The matrix M is CN-realizable. For the transformation C, convert all the white edges highlighted in pink to black edges B3 5B4 6B4 5 once, move them above the yellow- and blue-colored edges by ladder moves, and convert them again to the white edges. Definition 20. We …
Figure 19
Figure 19. Figure 19: The configurations a1 to a4. example, let M be a T0 upper triangular (0, 2)-matrix that has the configuration a3. Divide M into three matrices as M =         0 ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         +       …
Figure 20
Figure 20. Figure 20: The confugurations a3 and a4 are CN-realizable configurations. Un￾hook the 1st and 6th strands if the (1, 6) entry is 0. The formations shown in Section 4 are useful for finding further CN-realizable configurations. Example 6. The configurations b1, b2, b3 shown in […
Figure 21
Figure 21. Figure 21: The configurations b1, b2, b3. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: The CN-realizable configurations c1 to c26 that are derived from the snake formation. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: The CN-realizable configurations c27 to c44 that are derived from the snake formation. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: The CN-realizable configurations c45 to c64 that are derived from the snake formation. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 25
Figure 25. Figure 25: The CN-realizable configurations c65 to c79 that are derived from the hang-glider and snake formation. The configurations c65 to c70 include the H(1, 6; 2)-fromation. The configurations c71 to c74 include the H(1, 6; 3)- formation. The configurations c75, c76 include …
Figure 26
Figure 26. Figure 26: The CN-realizable configurations c80 to c100 that are derived from the loupe and snake formations. The configurations c80 to c84 include the L1(1, 6)-fromation. The configurations c85 to c90 include the L1(1, 5)-formation. The configurations c91 to c96 include the L1(…
Figure 27
Figure 27. Figure 27: The CN-realizable configurations d1 to d10 that are derived by the ladder diagram. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
Figure 28
Figure 28. Figure 28: The CN-realizable configurations d11 to d21 that are derived by the ladder diagram. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_28.png]

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Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [6]

    Kawauchi, Lectures on knot theory (in Japanese), Kyoritsu Shuppan Co

    A. Kawauchi, Lectures on knot theory (in Japanese), Kyoritsu Shuppan Co. Ltd, 2007

    work page 2007

  2. [1]

    Artin, Theory of braids, Ann

    E. Artin, Theory of braids, Ann. Math. 48 (1947), 101--126

    work page 1947

  3. [2]

    Burillo, M

    J. Burillo, M. Gutierrez, S. Krsti\' c and Z. Nitecki, Crossing matrices and Thurston's normal form for braids, Topl. Appl. 118 (2002), 293--308

    work page 2002

  4. [3]

    Elrifai, H

    E. Elrifai, H. Morton, Algorithms for positive braids, Quart. J. Math. Oxford 45 (2)(1994), 479 497

    work page 1994

  5. [4]

    F. A. Garside, The braid group and other groups, Quart J. Math. Oxford Ser. 20 (1969), 235–254

    work page 1969

  6. [5]

    Crossing Matrices of Positive Braids

    M. Gutierrez and Z. Nitecki, Crossing matrix of positive braids, arXiv:1805.12189

    work page Pith review arXiv

  7. [7]

    Shimizu, A

    A. Shimizu, A. Gill and S. Joshi, A note on the unknotting number and the region nuknotting number of weaving knots, arXiv:2408.14938

    work page arXiv

  8. [8]

    Characterization of the OU matrix of a braid diagram

    A. Shimizu and Y. Yaguchi, Characterization of the OU matrix of a braid diagram, to appear in Topol. Appl. (arXiv:2502.16035)

    work page Pith review arXiv

Show all 10 references
  1. [9]

    Shimizu and Y

    A. Shimizu and Y. Yaguchi, Determinant of the OU matrix of a braid diagram, J. Knot Theory Ramifications 34 (2025), 2550005

    2025

  2. [10]

    Thurston, Finite state algorithms for the braid group: Chapter 9 of Word processing in groups, D

    W. Thurston, Finite state algorithms for the braid group: Chapter 9 of Word processing in groups, D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Patterson and W. P. Thurston. Jones and Bartlett, Boston and London 1992

    1992

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