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REVIEW 3 major objections 7 minor 39 references

Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Jones polynomial can be localized to open curve segments with proven stability bounds.

desk verdict New localized Jones-polynomial descriptors look useful, but the persistence-stability theorem rests on a false premise and an unproved lemma; worth refereeing only if the stability proofs are fixed. read the letter →

arxiv 2411.17331 v1 pith:PLDMFZKJ submitted 2024-11-26 math.GT q-bio.BM

Ruzhi Song , Fengling Li , Jie Wu , Fengchun Lei , Guo-Wei Wei This is my paper · ORCID
classification math.GTq-bio.BM MSC 57K1492C10
keywords knotdataanalysiscurveJonespolynomiallocalizationstabilityproteinflexibilitypersistentbarcodeweightedpersistencediagram
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 Jones polynomial is a global knot invariant: it records the entanglement of a closed curve as a whole. This paper builds two local versions for collections of open or closed curves in 3-space by first cutting the curves into short segments and then evaluating the Jones polynomial at $t=10$. The multiscale Jones polynomial is an $n$-by-$m$ matrix whose $(i,k)$-entry is the Jones polynomial value of the set of segments within distance $[r_k,R_k)$ of segment $i$. The persistent Jones polynomial is a weighted barcode built from the Vietoris-Rips filtration of the segment distance matrix, with each facet weighted by the Jones polynomial of the corresponding segments. The central stability claim is that if a continuous map moves the curves by less than $\epsilon$ in the sup norm, then the matrix entries change by less than $\epsilon_J$ and the weighted bottleneck distance between the two persistent diagrams is less than $\max\{2\epsilon,\epsilon_J\}$, which is what real-world curve data requires.

What carries the argument

Two objects carry the argument. The first is the evaluation map $t\mapsto J(10)$ applied to the projection-averaged Jones polynomial of curve collections, which turns a polynomial invariant into a real-valued continuous function of the curve coordinates; this gives the bound $\epsilon_J$. The second is the facet barcode of a Vietoris-Rips filtration, the nested family of simplicial complexes built by connecting curve segments whose pairwise distances fall below a growing threshold; maximal simplices, or facets, are born and die as the threshold grows, and each facet is assigned the Jones polynomial of the corresponding curve segments as a weight. The stability proof couples these with a box-counting lemma from persistence-diagram theory, which bounds the number of points that a shifted diagram can place in a shrunk box, and with a matching argument that pairs each facet of the original diagram to a facet of the perturbed diagram.

What would settle it

Search over small configurations of three or four short curve segments: perturb one segment by a sup-norm shift of size $\epsilon$, compute the two weighted facet diagrams of the persistent Jones polynomial at $t=10$, and compare them with the weighted bottleneck distance. If any example gives a distance larger than $\max\{2\epsilon,\epsilon_J\}$, the theorem's bound is false. A simpler diagnostic is to find a configuration in which a facet dies at $y$ although one outside segment lies closer than $y$ to one facet member and another lies farther than $y$, which would contradict the proof's all-outside-segments distance premise.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Jones polynomial, a topological invariant that is normally defined for closed knots, can be localized to the scale of curve segments while retaining a controlled response to perturbation. Given a segmentation $P_n=\{l_1,\dots,l_n\}$ of a curve collection $L$, the multiscale model forms, for every segment $l_i$ and every distance window $[r,R)$, the set of segments within that window and evaluates the Jones polynomial of that subset at $t=10$. The persistent model filters the segments by the Vietoris-Rips complex and records the birth and death of each maximal simplex (facet), weighting the point $(x,y)$ in the resulting diagram by the Jones polynomial, again at $t=10$, of the segments that form the facet. Theorem 4.1 and Theorem 4.2 assert that both models are stable: under a continuous map $f$ with $\|f(L)-L\|_\infty<\epsilon$, the matrix entries move by less than $\epsilon_J$, and the weighted bottleneck distance between the diagrams is less than $\max\{2\epsilon,\epsilon_J\}$.

Load-bearing premise

The persistent stability proof assumes that a facet dies only when every outside segment is at least the death distance from every inside segment, whereas the Vietoris-Rips complex only guarantees this for some inside segment per outside segment; if that stronger distance condition fails, the diagram-matching argument in Section 4.2.2 has no support.

Editorial extensions

If this is right

  • The multiscale Jones polynomial yields a real-valued characteristic matrix for any segmented curve collection, so it can be fed directly into regression or machine-learning models; the paper reports B-factor correlations of $0.899$, $0.808$, and $0.720$ on small, medium, and large protein sets.
  • The persistent Jones polynomial produces weighted barcodes that distinguish protein secondary structures; the reported $\alpha$-helix weights span $-86$ to $0$ while the $\beta$-sheet weights span $-6$ to $0$.
  • The stability bounds imply that feature vectors built from either model are insensitive to small geometric noise, provided the segmentation and distance windows are held fixed.
  • Both models reduce to global knot information when a segment covers the whole curve and to trivial local information when segments are very short, so segmentation length is a tunable resolution parameter.

Reading between the lines

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

  • Editorial extension: the same segmentation-and-weighting construction could be applied to other knot polynomial invariants that vary continuously under small curve moves, giving analogous multiscale and persistent descriptors with their own stability bounds.
  • Editorial extension: because the facet barcodes track maximal simplices rather than homology classes, the paper opens a separate stability question for facet persistence itself; a matching lemma proved directly for Vietoris-Rips facets would either repair or replace the imported counting lemma.
  • Editorial extension: the evaluation point $t=10$ is a modeling choice; evaluating at several values of $t$ and concatenating the resulting matrices or diagrams would likely produce richer features without changing the stability argument.
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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

3 major / 7 minor

Summary. The paper proposes two localizations of the Jones polynomial for collections of disjoint open or closed curves in R^3, building on the averaged bracket polynomial of Barkataki and Panagiotou [5]. The multiscale Jones polynomial is an n x m matrix of evaluations at t=10, where each entry is the Jones polynomial of the set of segments within a distance window [r,R) of a given segment. The persistent Jones polynomial is defined by weighting the facets of a Vietoris-Rips filtration of the segmentation by the Jones polynomial of the corresponding curve segments, yielding weighted barcodes and weighted persistence diagrams. Section 4 claims stability: Theorem 4.1 states that the multiscale characteristic matrix changes by less than epsilon_J under a sup-norm perturbation of size epsilon, and Theorem 4.2 states that the weighted bottleneck distance between persistent Jones diagrams is less than max{2 epsilon, epsilon_J}. Section 5 applies the multiscale model to B-factor prediction and the persistent model to alpha-helix and beta-sheet barcodes.

Significance. If the stability theorems were valid, the paper would provide a useful new class of localized, stable knot-theoretic descriptors for curve data, extending the mGLI approach [35] and the weighted-barcode idea [8] to the Jones polynomial. The novelty lies in replacing global knot invariants with scale-dependent and persistence-based versions and in the reported B-factor correlation coefficients. The manuscript should be credited for building on prior definitions rather than inventing ad hoc entities. However, the theoretical contribution is the central claim, and it is not established: Theorem 4.2 relies on an unproved Box Lemma and a false death-time statement, and Theorem 4.1 relies on an unjustified boundary-separation step. The application results, while suggestive, are presented without statistical detail. Thus the paper cannot be accepted in its current form.

major comments (3)
  1. [§4.2.2, Lemma 4.1] Lemma 4.1 is imported from Cohen-Steiner-Edelsbrunner-Harer [9], but that result applies to persistence diagrams arising from homology persistence modules. Here the diagram D(Pn) is built from facets of a Vietoris-Rips filtration, and the quantity beta^y_x is a facet count, not a homology rank. No proof is given that the Box-Lemma inequality #(D(Pn) intersect R_{2 epsilon}) <= #(D(f(Pn)) intersect R) holds for these facet diagrams. Every counting step in the proof of Theorem 4.2 (the inequalities mu <= #(...), the equality #(...)=mu, and the exclusion of unmatched points) depends on this lemma. The persistent stability bound is therefore unsupported as written.
  2. [§4.2.2, death condition in Theorem 4.2] The proof of Theorem 4.2 states that if a facet Pn(p) dies at y, then any segment l outside Pn(p) is at distance at least y from every segment in Pn(p). In the Vietoris-Rips filtration this is false: a facet dies at the infimum over outside vertices of their maximum distance to the facet's vertices, so for each outside segment it is only guaranteed that some vertex of the facet is at distance at least y, not all. The lower bound on the death of the image facet under f is therefore not justified by the stated hypothesis. The proof also does not justify the upper bound 'death <= y'; under a perturbation the death time can increase by up to 2 epsilon. These issues affect the claimed bound ||f(p)-p||_infinity < 2 epsilon and the matching argument.
  3. [§4.1, Proposition 4.1] The proof of Proposition 4.1 concludes P^i_{r-4 epsilon, R+4 epsilon} = P^i_{r,R} from smallness of epsilon. This equality is false without a separation condition: if some distance d(l_i,l_j) equals r or R, or lies within 4 epsilon of the boundary, a perturbation of size < epsilon can move the distance across the boundary and change membership of the segment. Theorem 4.1, which relies on this equality to identify f(P^i_{r,R}) with f(P)^i_{r,R}, therefore does not follow from the hypotheses as stated. A non-degeneracy condition on the distance matrix relative to the radius parameters, or a reformulation of the stability statement, is needed.
minor comments (7)
  1. [Title] The title in the full text reads 'Multisacle Jones Polynomial'; the typo should be corrected.
  2. [§3.1] The distance d(l_i,l_j) is described as 'the upper bound of the Eulerian distances'; it should be explicitly defined as sup_{x in l_i, y in l_j} |x-y| to avoid ambiguity.
  3. [§4.2] The multiplicity mu^i_j is defined for a 'fixed integer k' without first specifying the range of facet dimensions; the dependence of D(Pn) on k should be stated.
  4. [§4.2.1] The weighted bottleneck distance treats diagonal points as having infinite multiplicity in the diagrams, but the matching definition for multi-sets of pairs does not specify how diagonal points are handled; the diagonal convention should be made explicit.
  5. [§5.1] The reported correlation coefficients (0.899, 0.808, 0.720) are averages across protein sets, but no error bars, number of proteins per set, or statistical comparison with the listed baselines is provided.
  6. [§5.2] The statement that 'a greater color difference between two bars suggests a greater difference between the sets of curve segments they represent' is not a quantitative claim; the color scale is the Jones polynomial weight and the relation to entanglement complexity is unexplained.
  7. [§4.2.2] The final sentence of Section 4.2.2 says the diagrams are stable under 'possibly irregular perturbations'; the term 'irregular' is not defined in the paper.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the stability claims are derived from the externally quoted continuity of the open-curve Jones polynomial and from explicit perturbation bounds, not from the conclusions being proved; the cited Box Lemma and the facet-death condition are correctness gaps, not circular reductions.

full rationale

The derivation chain for Theorem 4.1 begins with Proposition 2.1, quoted from [5], that the open-curve Jones polynomial is a continuous function of the curve coordinates, and then bounds each characteristic-matrix entry by |J_{P^i}(10) - J_{f(P^i)}(10)| < ε_J together with the segment-set equality of Proposition 4.1. This is a direct input-to-output estimate, not a restatement of the theorem. The persistent stability theorem is likewise assembled from explicit bounds: births move by at most 2ε via the Rips distance inequality, deaths are claimed to move by at most 2ε, and weights move by at most ε_J by the same continuity result, so the bottleneck bound max{2ε, ε_J} follows from the constructed matching. No fitted parameter is renamed as a prediction, and the target inequality is not assumed among the hypotheses. The main weaknesses are mathematical rather than circular: Lemma 4.1 from [9] is a homology persistence-diagram statement and the manuscript supplies no proof that the facet-based diagrams satisfy it, and the assertion in Section 4.2.2 that an outside segment is at distance at least y from every segment of the facet is stronger than the Rips death condition, which only requires the maximum distance over the facet to be at least y for each outside segment. These are unproved-step and correctness risks that a referee should ask the authors to repair, but they do not make the theorem definitionally equivalent to its input. The self-citations to [35] and [8] are contextual or comparative: the mGLI model is used as a benchmark, and the weighted barcode is used as a representational device; neither supplies the stability bound. Hence the paper is not circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central stability claims rely mostly on standard metric facts and on prior results from [5] and [9]. The multiscale theorem additionally needs a boundary-separation condition for finite sets of distances (not stated). The persistent theorem needs an extension of the Box Lemma to facet diagrams and uses a false geometric assertion about facet death. The application section introduces hand-chosen parameters (t=10, radii grid, Lasso coefficient) that are not derived.

free parameters (5)
  • Jones polynomial evaluation point = t = 10
    Chosen to convert polynomial weights to real numbers for matrices and barcodes (Sections 3.1, 3.2). Stability is stated only for this specialization.
  • Distance window radii r and R = r in [4,15] Angstrom step 0.25; R = r+1 Angstrom
    Used in B-factor application; based on average C-alpha to C-alpha distance of 3.8 Angstrom, and no sensitivity analysis is reported.
  • Lasso regularization parameter = 0.16
    Chosen to avoid overfitting in the B-factor regression; no tuning procedure or cross-validation is described.
  • Segmentation P_n = cut at midpoints between consecutive C-alpha atoms
    Both models depend on the segmentation; the paper acknowledges the choice is critical (Remark 3.1, Section 6).
  • Distance between segments = distance between C-alpha atoms within segments
    The distance d(li,lj) is application-defined; the stability theorems allow any metric, but the B-factor implementation fixes this choice.
assumptions (5)
  • domain assumption The open-curve Jones polynomial from [5] is well-defined, has real coefficients, and is continuous in the curve coordinates.
    Invoked in Remark 4.1 and Theorem 4.1 to bound changes in J(10) under perturbations; quoted from Proposition 2.1 without independent proof in this paper.
  • ad hoc to paper The Cohen-Steiner-Edelsbrunner-Harer Box Lemma applies to the facet-based persistence diagrams defined in Section 3.2.
    Cited as Lemma 4.1 in Section 4.2.2, but [9] proves the lemma for homology persistence diagrams, not for diagrams of facets; no proof is supplied here.
  • ad hoc to paper For a facet that dies at y, every outside segment is at distance at least y from every segment in the facet.
    Used in the proof of Theorem 4.2 to locate images of points; the correct Vietoris-Rips condition is that each outside segment has at least one facet segment at distance at least y.
  • standard math Perturbing each point by at most epsilon changes any pairwise distance by at most 2 epsilon.
    Used repeatedly in Proposition 4.1 and Theorem 4.2; follows from the triangle inequality.
  • standard math A finite Vietoris-Rips filtration has finitely many critical values and its final complex is an (n-1)-simplex.
    Remark 3.2 and the definition of the filtration in Section 3.2; standard for finite metric spaces.

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Pith. "Pith review of Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis." pith.science (2026). https://pith.science/paper/PLDMFZKJ

@misc{pith2026241117331,
  author       = {Pith},
  title        = {Pith review of: Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLDMFZKJ}},
  note         = {Machine review of arXiv:2411.17331}
}
read the original abstract

Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory primarily focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial, namely the multiscale Jones polynomial and the persistent Jones polynomial, are proposed. The stability of these models, especially the insensitivity of the multiscale and persistent Jones polynomial models to small perturbations in curve collections, is analyzed, thus ensuring their robustness for real-world applications.

Figures

Figures reproduced from arXiv: 2411.17331 by the authors.

Figure 1
Figure 1. Representation of the segment cycle of a ∈ G. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Hopf linkoid and its states 1 Sˆ 1(1) = 3 Sˆ 1(4) = 2 Lˆ(3) = 4 (a) Sˆ 1 = (1, 3)(2, 4) 3 Sˆ 2(3) = 4 1 Sˆ 2(1) = 2 (b) Sˆ 2 = (1, 2)(3, 4) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Segment cycles of two states 2.2 Jones polynomial The bracket polynomial of linkoids in S 2 or R 2 is defined through an extension of the bracket polynomial of links. The following initial conditions and diagrammatic relations are sufficient for the skein computation of the bracket polynomial of linkoids. Definition 2.1. Let L be a linkoid diagram with n components. The bracket polynomial of the linkoid is uniquely … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The multiplicity of the point (ri , rj ) is the alternating sum at the corners of the lower right square. When adding other multiplicities, cancellations between plus and minus signs occur. Definition 4.1. The diagram D(Pn) ⊂ R¯ 2 of the persistent Jones polynomial for…
Figure 5
Figure 5. Figure 5: The shaded squares are centered at the black points of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Comparison of B-factor predictions on three benchmark datasets between our Jones [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Comparison of B-factor values for protein [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: An alpha helix (left) and its persistent Jones polynomial barcodes (right). [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: A beta sheet (left) and its persistent Jones polynomial barcodes (right). [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

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    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.