Pith. sign in

REVIEW 2 major objections 2 minor 9 references

Linking number of grid models

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

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

Pith's one-line read For a random two-component grid link, the $u$-th moment of the linking number is a polynomial in the size $n$ of degree at most $u$; all odd moments vanish, and the normalized linking number converges weakly to a unique distribution.

desk verdict Solid new enumeration for grid-model linking number, with a real but fixable gap in the weak-convergence proof. read the letter →

arxiv 2506.02369 v1 pith:KQVKOUUU submitted 2025-06-03 math.GT

Senja Barthel , Yuka Kotorii This is my paper
classification math.GT MSC 57M2557M27
keywords randomlinkgriddiagramlinkingnumbermomentpolynomialweakconvergenceknotinvariantfinite-type
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

Two-component links in the grid model are specified by two independent, uniformly random permutations of size $2n$, making the linking number a random variable on grid diagrams. The paper's central claim is that for a fixed moment order $u$, the $u$-th moment $\mathbb{E}[\operatorname{lk}(L_n)^u]$ is a polynomial in $n$ of degree at most $u$, with all odd moments equal to zero. This pins down how the invariant grows: the leading term of the second moment is $n^2/36$, so a typical random grid link has linking number on the order of $n$. Using the moment-polynomial structure, the paper proves that the normalized linking number $\operatorname{lk}(L_n)/n$ converges weakly to a unique limiting distribution.

What carries the argument

The load-bearing object is the type of an index sequence. A type records, for a length-$u$ sequence $(k_1,\dots,k_u)$ of row or column labels on $\{1,\dots,n\}$, which indices are equal, which are consecutive modulo $n$, and which are separated by at least two, organizing the sequence into chains of blocks. The counting lemmas show that the number of permutations $\sigma$ or $\pi$ satisfying a conjunction of crossing conditions depends only on the types of the two index sequences, and that the number of sequences of a given type $P$ with $s$ chains of lengths $l(p_h)$ is the falling factorial $n(n-1-\sum_h l(p_h))!/(n-s-\sum_h l(p_h))!$. This reduces the moment sum to a finite sum over types of $n$-independent combinatorial data, from which the polynomial degree bound follows.

What would settle it

Exhaustively enumerate all pairs $(\sigma,\pi)\in S_{2n}^2$ for small $n$, compute the exact second and fourth moments of $\operatorname{lk}$, and compare with the predicted polynomial form: degree at most $u$ and, for $u=2$, leading coefficient $1/36$. A more targeted check is to take two index sequences of the same type that start at different positions and count, for a fixed sign vector, how many permutations satisfy the conjunction of crossing conditions; unequal counts would disprove Lemma 4.8 and with it Theorem 1.1.

Watch

Extended reading notes

Core claim

Theorem 1.1 states that for a uniformly random 2-component oriented grid link $L_n$ of order $2n$, the moment $\mathbb{E}[\operatorname{lk}(L_n)^u]$ is a polynomial in $n$ of degree $d \le u$, and $\mathbb{E}[\operatorname{lk}(L_n)^u] = 0$ for odd $u$. The proof is combinatorial: the moment is expanded as a sum over $u$ pairs of row and column indices and over sign vectors, the number of permutation pairs realizing a given sign pattern is counted by classifying the index sequences into types, and a sign-cancellation lemma kills all types containing an isolated index. The surviving terms are polynomials in $n$, and for even $u$ the maximal degree $u$ is attained only when both index sequences split into $u/2$ blocks of size $2$. For $u=2$ this yields the explicit leading coefficient $1/36$, and Corollary 1.2 converts the resulting moment bounds into weak convergence of $\operatorname{lk}(L_n)/n$.

Load-bearing premise

The load-bearing premise is that the number of permutation pairs exhibiting a given crossing-sign pattern depends only on the type of the index sequences (Lemma 4.8), an assertion justified in the paper by a sketch rather than a full explicit bijection; if two index sequences of the same type could yield different counts, the polynomial-degree bound and the cancellations behind Theorem 1.1 would fail.

Editorial extensions

If this is right

  • The normalized linking number $\operatorname{lk}(L_n)/n$ has a well-defined limit distribution, and the moment estimates in the proof show that all moments of that limit are finite.
  • All odd moments of the limit are zero, and since the moments determine the distribution, the limiting law is symmetric about zero.
  • The variance of the linking number satisfies $\operatorname{Var}(\operatorname{lk}(L_n)) \sim n^2/36$, so the invariant fluctuates on scale $n$ as the grid grows.
  • For each even $u$, the leading coefficient of the $u$-th moment is a finite signed sum over maximal type splittings, so the full moment sequence of the limit distribution is in principle computable.
  • The result gives the grid-model analogue of the known petaluma-model weak convergence of the normalized linking number, supporting the broader expectation that normalized finite-type invariants of random links converge.

Reading between the lines

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

  • A next step the paper leaves implicit is to identify the limiting distribution; because each moment is a finite sum over type pairs, a symbolic computation for small $u$ could reveal whether the limit matches a familiar law such as a scaled symmetric distribution.
  • The type-counting argument is tailored to the uniform two-permutation grid model, but the same strategy of classifying coincidences and adjacencies could be tried on other statistics that decompose into sums of local crossing contributions, including invariants of links with more than two components.
  • A direct numerical check of the paper's variance prediction is feasible now: sampling grid diagrams for $n$ from about 10 to 100 should show the sample variance of $\operatorname{lk}/n$ approaching $1/36$, with a $1/n$ correction coming from the lower-degree polynomial terms.
  • If one computed the fourth moment's leading coefficient, comparing it with $3(1/36)^2$ would test whether the limiting distribution is Gaussian; the paper does not make that comparison, but its methods supply the ingredients.
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.

Referee Report

2 major / 2 minor

Summary. The paper studies the linking number of a uniformly random 2-component grid link diagram of order 2n, represented by two independent uniform permutations in S_{2n}. The authors prove that the uth moment of the linking number is a polynomial in n of degree at most u, with all odd moments zero, and they compute the leading coefficients for the second moment (limit 1/36 for E[(lk/n)^2]). They then claim that the normalized linking number lk/n converges weakly to the unique distribution with these limiting moments. The proof combines a combinatorial decomposition of crossing signs into 'types' of index sequences, counting formulas for permutations satisfying chains of crossing conditions, and a moment-method argument using a Carleman-type criterion.

Significance. If the results hold, the paper provides a parameter-free asymptotic description of a natural random-link model and offers a useful comparison with the petaluma model, where the analogous weak convergence was already known. The combinatorial framework—type classification, permutation counts, and factorial ratios—is potentially reusable for other finite-type invariants. The paper's strengths include a direct derivation of polynomiality from the model definition, an explicit second-moment coefficient, and a simple symmetry argument for vanishing odd moments. However, the proof of the weak-convergence corollary currently rests on a false numerical bound, so the advertised main corollary is not yet established by the arguments presented.

major comments (2)
  1. [Section 5, proof of Corollary 1.2] The displayed inequality bounding μ_{2u} is false as written. For u=1, using the values tabulated in Example 4.19, the middle sum Σ_{P,Q} 2^{2u} max_ε N_{P,Q,ε}/(n_{P,Q}!)^2 equals 79/150, while the claimed upper bound ((2u)!/(2^u u! 3^u))^2 2^{2u} equals 4/9. Moreover, the subsequent equality is algebraically incorrect: ((2u)!/(2^u u! 3^u))^2 2^{2u} simplifies to (2u)!^2/(u!^2 3^{2u}), not to (2u)!^2 3^{2u}/u!^2. Consequently, the limsup estimate needed for Theorem 5.1 is not proved. Since this is the only step converting the moment computation into weak convergence, Corollary 1.2 is currently unsupported; a corrected bound (likely via a sharper combinatorial estimate) is required.
  2. [Section 4, Lemma 4.8] The proof of type-invariance of the permutation counts is only sketched with 'one can check' and does not exhibit a bijection between the sets {σ : ⋀_i A_{k_i,l_i}} and {σ : ⋀_i A_{k'_i,l_i}} for sequences k and k' of the same type P. Since Lemma 4.8 justifies the reduction from (4.2) to (4.3) and the subsequent summation over types, the argument is load-bearing for Theorem 1.1. The paper should supply a complete proof or a precise bijection for this lemma.
minor comments (2)
  1. [Section 5, first paragraph of the proof] The phrase 'μ_u be the constant term of n of uth moment of lk(Ln)/n' is imprecise; μ_u should be defined explicitly as the limit lim_{n→∞} E[(lk/n)^u], equivalently the coefficient of n^u in the polynomial E[lk^u].
  2. [Throughout] There are many typographical issues and OCR artifacts (e.g., broken words like 'i n', 'the the', and unclear subscripts/superscripts in A_{k_i,l_i} and B_{k_i,l_i}). A careful proofreading and consistent notation display would significantly improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the moment bound and weak convergence are derived from the model definition using direct counting and Durrett's external moment-convergence theorem; nothing is fitted or imported from the authors' prior work.

full rationale

The derivation starts from the definition of the grid model (two independent uniform permutations sigma, pi) and the sign formula for the linking number in Proposition 3.1. Lemma 4.2 expands the u-th moment as a sum over index sequences and sign vectors, and Lemmas 4.3-4.4 decompose the crossing conditions into independent conditions on sigma and pi. Lemma 4.8 asserts that the count depends only on the type of the index sequence; this step is sketched with 'one can check' rather than fully proved, but it does not assume the conclusion of Theorem 1.1. Lemma 4.12 counts sequences of a given type directly, and Lemma 4.13 cancels contributions from length-one types. Substituting these counts into equation (4.3) yields a polynomial in n of degree at most u, with the leading coefficient formula in Remark 4.18 and the u=2 value 1/36 in Example 4.19. No parameter is fitted to data, no result by the same authors is used as an input, and the only external ingredient is the standard moment-convergence criterion cited as Durrett [5]. A skeptical check of the inequality in the proof of Corollary 1.2 may indicate a correctness gap or algebraic slip, but that is a proof issue rather than circularity: the target weak-convergence statement is not assumed as an input. There is therefore no circular dependence in the paper's derivation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the uniform random permutation model and on exact counting of permutation patterns; no free parameters are fitted and no new physical entities are introduced. The combinatorial 'types' of Definition 4.5 are internal bookkeeping, not entities with independent evidence.

assumptions (4)
  • domain assumption Uniform independent random permutations define the random link model
    Section 2 defines L_n by drawing σ and π independently and uniformly from S_{2n}; all moment statements are taken with respect to this measure.
  • domain assumption Grid diagrams are universal for links
    Introduction cites Cromwell [4] for the fact that every link admits a grid diagram, justifying the model.
  • standard math Durrett's moment convergence criterion
    Theorem 5.1 [5] is used in Corollary 1.2 to pass from moment limits to weak convergence.
  • domain assumption Fixed moment order u is smaller than n
    Lemmas 4.2, 4.5 and 4.12 require u<n; for fixed u the asymptotic statements let n tend to infinity, so the condition is eventually satisfied.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linking number of grid models." pith.science (2026). https://pith.science/paper/KQVKOUUU

@misc{pith2026250602369,
  author       = {Pith},
  title        = {Pith review of: Linking number of grid models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQVKOUUU}},
  note         = {Machine review of arXiv:2506.02369}
}
abstract

This paper studies the linking numbers of random links within the grid model. The linking number is treated as a random variable on the isotopy classes of 2-component links, with the paper exploring its asymptotic growth as the diagram size increases. The main result is that the $u$th moment of the linking number for a random link is a polynomial in the grid size with degree $d\leq u$, and all odd moments vanishing. The limits of the moments of the normalized linking number are computed, and it is shown that the distribution of the normalized linking number converges weakly as the grid size tends to infinity.

Figures

Figures reproduced from arXiv: 2506.02369 by the authors.

Figure 1
Figure 1. The 2-component oriented grid link diagram L (1,3,2,4),(2,4,1,3) 2 of order 4. The link has linking number +1. A random knot in the grid model is obtained by choosing σ and π independently and uniformly at random [6]. Similarly, a 2-component oriented random link Ln in the grid model is defined by choosing σ and π independently and uniformly at random. For 2-component random links in the the petaluma model, the limi… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Adams, T

    C. Adams, T. Crawford, B. DeMeo, M. Landry, A.T. Lin, M.K. Montee, S. Park, S. Venkatesh and F. Yhee, Knot projections with a single multi-crossing , Journal of Knot Theory and Its Ramifications 24 , 03 (2015), 1550011

    work page 2015

  2. [2]

    Brunn, \" U ber verknotete Kurven ,

    H. Brunn, \" U ber verknotete Kurven , . (1897), pp. 256--259

    work page

  3. [3]

    U ber verknotete Kurven , In Verhandlungen des ersten Intemationalen Mathematiker Kongresses, Z\

    H. Brunn, \" U ber verknotete Kurven , In Verhandlungen des ersten Intemationalen Mathematiker Kongresses, Z\" u rich 1897 (1898), 256--259

    work page

  4. [4]

    Cromwell, Arc presentations of knots and links , Banach Center Publications 42 (1998), 57--64

    P.R. Cromwell, Arc presentations of knots and links , Banach Center Publications 42 (1998), 57--64

    work page 1998

  5. [5]

    Cromwell, Embedding knots and links in an open book I: Basic properties , Mathematical Proceedings of the Cambridge Philosophical Society 119 , 2 (1996), 309

    P.R. Cromwell, Embedding knots and links in an open book I: Basic properties , Mathematical Proceedings of the Cambridge Philosophical Society 119 , 2 (1996), 309

    work page 1996

  6. [6]

    Durrett, Probability: Theory and Examples , 5th edition, Cambridge Series in Statistical and Probabilistic Mathematics, 49, Cambridge University Press, Cambridge, UK (2019)

    R. Durrett, Probability: Theory and Examples , 5th edition, Cambridge Series in Statistical and Probabilistic Mathematics, 49, Cambridge University Press, Cambridge, UK (2019)

    work page 2019

  7. [7]

    Even-Zohar, J

    C. Even-Zohar, J. Hass, N. Linial and T. Nowik Invariants of random knots and links , Discrete Computational Geometry 56 (2016), 274--314

    work page 2016

  8. [8]

    Even-Zohar, J

    C. Even-Zohar, J. Hass, N. Linial and T. Nowik The distribution of knots in the Petaluma model , Algebraic & Geometric Topology 18 (2018) 3647–-3667

    work page 2018

Show all 9 references
  1. [9]

    Farhi, D

    E. Farhi, D. Gosset, A. Hassidim, A. Lutomirski, P. Shir, Quantum money from knots , In Proceedings of the 3nd Innovations in Theoretical Computer science Conference, Association for Computing Machinery, New York, NY, USA (2012), 276--289

    2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.