{"id":"23007cf2-0c39-4675-8c9a-525a83250c1c","arxiv_id":"2411.17331","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"Two localized, stable versions of the Jones polynomial are introduced for analyzing local and global entanglement of open curves, with applications to protein B-factor prediction.","lead":"This paper builds two local versions of the Jones polynomial, a knot-theory invariant, so it can measure entanglement of open curves like protein backbones at different scales. The authors show the descriptors are stable under small shape changes and use them to predict protein flexibility, with reported accuracy that beats earlier methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Persistent stability theorem is unproven: Lemma 4.1 imports the homology Box Lemma to facet diagrams without justification, and the facet-death condition in §4.2.2 is misstated, so the headline bound dB < max{2ε, ε_J} is not established.","rationale":"The reader's weakest assumption correctly identifies that Theorem 4.2 rests on unjustified premises. I agree that the proof is invalid and that the persistent stability claim is not established. However, I would not place the full weight on the false universal distance condition, because the correct existential condition follows directly from the definition of facet death and would repair that particular step. The more fundamental problem is the unproved applicability of Lemma 4.1: facet diagrams of Rips complexes are not homology persistence modules, and the manuscript supplies no proof of the Box Lemma for them. This is not a minor typo; it is the core of the stability argument. The multiscale stability theorem (Theorem 4.1) is more plausible, though its proof also needs a generic-distance caveat for Proposition 4.1. The application section is additionally non-reproducible: no code, no data, and hyperparameters are fixed without validation, which prevents independent verification of the claimed B-factor results. These issues justify the reader's REJECT verdict: the central claim is unproven as written, and the manuscript cannot be accepted without a corrected proof and reproducible experiments. I would not move the verdict to CONDITIONAL because the gap concerns the main theorem, not a peripheral detail, and the theorem may or may not survive a correct analysis.","tokens_in":17502,"tokens_out":28124,"duration_ms":274152,"concrete_test":"Run a randomized computational test on small point configurations: for n = 4 to 8 random point sets L in the unit square, compute the Rips facet diagram D(L) by enumerating maximal cliques over a fine scale grid, apply a random perturbation f with sup norm ε, recompute D(f(L)), and verify the Box Lemma inequality #(D(L) ∩ R_{2ε}) ≤ #(D(f(L)) ∩ R) for all grid boxes R, as well as the final bound dB ≤ max{2ε, ε_J}. A single violation would refute Theorem 4.2 as stated; if thousands of trials show no violation, the claim is plausible but the submitted proof would still need a correct derivation instead of the imported Lemma 4.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.2, the paper's central quantitative stability claim for the persistent Jones polynomial, is not proven as written. The proof in §4.2.2 relies on Lemma 4.1, a verbatim citation of the Cohen–Steiner–Edelsbrunner–Harer Box Lemma. That lemma is a statement about homology persistence modules and their rank functions; the manuscript gives no argument that facet diagrams (maximal cliques of a Vietoris–Rips filtration) satisfy the required inequalities, so the lemma is inapplicable as stated. Independently, the proof asserts that if a facet dies at y, then every outside segment is at distance at least y from every segment of the facet. This is false: in a Rips complex the death time y is the minimum, over outside segments w, of max_{v in facet} d(w,v); for a given w only one facet segment needs to be at distance at least y. This error appears immediately after the definition of δ_L and is used to bound the image facet's death, so the matching argument does not go through. Even though the false universal condition can likely be replaced by the correct existential one, the unproved Box Lemma remains a genuine gap: the facet diagram is not a homology persistence diagram, and no stability theorem for such diagrams is supplied. Consequently the persistent stability bound is unsupported, and this is load-bearing because it is one of the two advertised main results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":36,"tokens_out":12676,"duration_ms":245293,"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":[{"comment":"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.","section":"§4.2.2, Lemma 4.1"},{"comment":"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.","section":"§4.2.2, death condition in Theorem 4.2"},{"comment":"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.","section":"§4.1, Proposition 4.1"}],"minor_comments":[{"comment":"The title in the full text reads 'Multisacle Jones Polynomial'; the typo should be corrected.","section":"Title"},{"comment":"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.","section":"§3.1"},{"comment":"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.","section":"§4.2"},{"comment":"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.","section":"§4.2.1"},{"comment":"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.","section":"§5.1"},{"comment":"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.","section":"§5.2"},{"comment":"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.","section":"§4.2.2"}],"recommendation":"reject","confidential_remarks":"The main theoretical results are not established: the false death-time statement and the unproved Box Lemma are not local presentation issues but load-bearing gaps in the advertised stability theorems. The application section might support an application-focused paper if the stability claims were removed or replaced with a correctly proved statement, but the present manuscript cannot be accepted without substantial new mathematics or a redefinition of the persistent invariant."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper introduces two genuinely new local descriptors built from the Jones polynomial for open curves: a multiscale matrix using distance windows, and a facet-based persistence diagram weighted by Jones polynomial values. That part is well motivated and clearly explained. The idea of localizing an otherwise global knot invariant by cutting curves into segments is natural and worth having in the literature. The B-factor numbers look plausible, though the manuscript gives no code, no data, and no error bars, so the empirical claim isn't checkable from the paper.\n\nThe soft spot is load-bearing. The persistent stability theorem, Theorem 4.2, is not proven as written. The proof asserts that when a facet dies at y, every outside segment is at distance at least y from every segment of the facet. In a Vietoris-Rips complex that's false: it should be that for each outside segment there is some facet segment at distance at least y. The paper then uses the universal version to bound the death of the image facet, so the matching argument doesn't go through. Also, Lemma 4.1 imports the Cohen-Steiner-Edelsbrunner-Harer Box Lemma, which is a statement about homology persistence modules, and applies it to facet diagrams without proof. Facet counts are not homology ranks; there is no established stability theorem for them. So the advertised bound dB < max{2ε, ε_J} is unsupported.\n\nTheorem 4.1 has a smaller but real problem: Proposition 4.1 effectively assumes the window boundaries r and R avoid all pairwise distances by a margin of at least 4ε. That's not stated, so the theorem's generality is overclaimed.\n\nThe math is not hopeless. The existential reading of facet death likely gives a Lipschitz bound with slightly different constants, and the multiscale argument can be fixed by adding a separation condition. But the paper as it stands doesn't deliver the stable persistence claim it advertises.\n\nThe right reader is someone working on topological descriptors for curve data—this is a serious attempt to bring the Jones polynomial into data analysis, and the writing is honest about prior work. I'd send it to peer review because the constructions and the application are interesting enough to warrant referee time, but it needs major revision: fix or restate the two stability theorems, and provide reproducible code and data for the B-factor benchmark.","headline":"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.","tokens_in":18350,"tokens_out":4043,"would_cite":false,"duration_ms":39635,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K14","92C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Jones polynomial can be localized to open curve segments with proven stability bounds.","keywords":["knot data analysis","curve data analysis","Jones polynomial","localization","stability","protein flexibility","persistent barcode","weighted persistence diagram"],"falsifier":"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.","tokens_in":17271,"feed_emoji":"🪢","tokens_out":10572,"duration_ms":94177,"temperature":0.7,"pith_summary":"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.","feed_headline":"Knot polynomial goes local and stable for open curves","feed_subtitle":"Two new Jones-polynomial descriptors stay close under small perturbations and predict protein flexibility.","key_machinery":"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.","core_discovery":"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 $\\max\\{2\\epsilon,\\epsilon_J\\}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the underlying Jones polynomial for collections of open or closed curves averaged over projection directions, from which both new models are built.","marker":"[5]"},{"why":"Supplies the box lemma bounding diagram-point counts, the key tool used in the proof of Theorem 4.2.","marker":"[9]"},{"why":"Introduces the multiscale Gauss link integral, the prior knot-data-analysis method that the new models extend and outperform on B-factor prediction.","marker":"[35]"},{"why":"Provides the three protein benchmark datasets used for the B-factor prediction comparison.","marker":"[30]"},{"why":"Justifies the continuity of knot-polynomial evaluations under small curve perturbations, invoked as Remark 4.1.","marker":"[26]"},{"why":"Defines the classical Jones polynomial that the paper localizes.","marker":"[16]"},{"why":"Introduces weighted barcodes, the representation used for the persistent Jones polynomial.","marker":"[8]"}],"fun_headline_variants":["Local Jones polynomials for open 3D curves","Stable Jones invariants for noisy curve data","Multiscale and persistent knot descriptors","Jones polynomial tamed for local curve features","Persistent knot data via Jones polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Local Jones polynomials for open 3D curves","Stable Jones invariants for noisy curve data","Multiscale and persistent knot descriptors","Jones polynomial tamed for local curve features","Persistent knot data via Jones polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000948,"raw_usage":{"total_tokens":4023,"prompt_tokens":896,"completion_tokens":3127,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":3061}},"tokens_in":512,"tokens_out":3127,"duration_ms":22331,"temperature":1.0,"reasoning_tokens":3061,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:16:11.189233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The jones polynomial of collections of open curves in 3-space.Proceedings of the Royal Society A, 478(2267):20220302, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the underlying Jones polynomial for collections of open or closed curves averaged over projection directions, from which both new models are built."},{"cited_title":"Stability of persistence diagrams","cited_arxiv_id":null,"evidence_quote":"Supplies the box lemma bounding diagram-point counts, the key tool used in the proof of Theorem 4.2."},{"cited_title":"Knot data analysis using multiscale gauss link integral.Proceedings of the National Academy of Sciences, 121(42):e2408431121, 2024","cited_arxiv_id":null,"evidence_quote":"Introduces the multiscale Gauss link integral, the prior knot-data-analysis method that the new models extend and outperform on B-factor prediction."},{"cited_title":"Coarse grained normal mode analysis vs","cited_arxiv_id":null,"evidence_quote":"Provides the three protein benchmark datasets used for the B-factor prediction comparison."},{"cited_title":"Knot polynomials of open and closed curves","cited_arxiv_id":null,"evidence_quote":"Justifies the continuity of knot-polynomial evaluations under small curve perturbations, invoked as Remark 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the classical Jones polynomial that the paper localizes."},{"cited_title":"Persistent cohomology for data with multicomponent heterogeneous information","cited_arxiv_id":null,"evidence_quote":"Introduces weighted barcodes, the representation used for the persistent Jones polynomial."}],"review_version":1}