Pith. sign in

REVIEW 5 major objections 4 minor 18 references

Profiles of Critical Flat Ribbon Knots

T0 review · 5 major / 4 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 This paper computes exact minimal ribbonlengths for standard flat diagrams of the Salomon knot ($8+2\pi$), the Turk's head knot ($16+3\pi$), the granny and square knots ($12+3\pi$), and two infinite families of ribbon knots.

desk verdict New exact ribbonlength values that would be a real advance if the proofs held, but the connected-sum additivity and lower-bound arguments are asserted, not established, so this needs major revision before the numbers can be trusted. read the letter →

arxiv 2506.04403 v1 pith:26RUYILG submitted 2025-06-04 math.GT

José Ayala This is my paper · ORCID
classification math.GT MSC 57K1049Q1053C42
keywords flatribbonknotsribbonlengthknotdiagramsdiskgeometrictheorySalomonTurk'sheadconnectedsums
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

Flat ribbon knots are constant-width strips folded in the plane, and ribbonlength is the ratio of strip length to width. Before this paper, the only knot diagrams whose minimal ribbonlength was known exactly were the standard unknot and the trefoil. Here the paper establishes exact minimal ribbonlengths for standard diagrams of the Salomon knot ($8+2\pi$), the Turk's head knot ($16+3\pi$), the granny and square knots ($12+3\pi$), connected sums of $n$ Salomon knots ($8n+2\pi$), and an infinite rectangular family ($\pi(m+n)+8mn$), and it proves a criticality theorem for another infinite Salomon family. These values give the first exact data points beyond the trefoil for a metric tabulation of knots.

What carries the argument

The load-bearing object is disk space $\mathcal{D}$, the space of finite collections of planar loops whose complementary regions each contain a unit disk and whose curves obey a weak separation bound. Length minimisers in $\mathcal{D}$ are cs curves, finite $C^1$ concatenations of unit-radius circular arcs and straight segments. Lemma 4.1 supplies the variational test: the derivative of core length under admissible disk-centre perturbations is a sum of unit tangent contributions, and a diagram is critical when this derivative vanishes, with blocked directions analysed by rolling one unit disk around another. The paper uses this criterion to run a geometric gradient descent from rectangular lattice diagrams and to verify that terminal configurations achieve all lower bounds, typically by recognising them as minimal-perimeter stadium curves enclosing unit disks.

What would settle it

An independent numerical minimisation of the granny knot's standard diagram that does not impose the connected-sum overlap rule would settle Corollary 4.5: a ribbon-valid configuration with ribbonlength below $12+3\pi$ falsifies it, and one that converges to $12+3\pi$ supports it.

Watch

Extended reading notes

Core claim

Working in the space of disk diagrams, the paper proves that the standard Salomon diagram has minimal ribbonlength $8+2\pi$, the standard Turk's head diagram has minimal ribbonlength $16+3\pi$, the standard granny and square diagrams both have minimal ribbonlength $12+3\pi$, and a connected sum of $n$ Salomon diagrams has minimal ribbonlength $8n+2\pi$. It also proves that every $m\times n$ rectangular lattice arrangement has minimal ribbonlength $\pi(m+n)+8mn$, and that the $n$-th element of the Salomon family is ribbonlength-critical with ribbonlength $8\sum_{k=1}^n k+(n+1)\pi$. The paper presents these as minima within the standard combinatorial type of each diagram, not as a result about all possible realisations of the same knot, and it explicitly withholds a minimality claim for the Salomon family because of known subtleties in extremal disk packings.

Load-bearing premise

The connected-sum arguments assume without proof that gluing two minimal ribbon diagrams subtracts exactly the perimeter of one unit disk from their combined length, and this unproved additivity rule carries the granny, square, and connected-sum Salomon values.

Editorial extensions

If this is right

  • The granny and square knots, although distinct as knots, share the exact minimal ribbonlength $12+3\pi$, so chirality does not affect the value for these standard diagrams.
  • The Salomon connected-sum formula $8n+2\pi$ gives an infinite family of composite knots with exactly known minimal ribbonlength, the first such family in flat ribbon knot theory.
  • The rectangular family provides an infinite array of diagrams with the closed-form bound $\pi(m+n)+8mn$, giving explicit test cases for the paper's conjecture that ribbonlength grows at most linearly in crossing number.
  • The critical Salomon family, although not minimal, yields diagrams whose ribbonlength cannot be decreased locally under the separation bound, giving explicit upper bounds for other knots and links.

Reading between the lines

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

  • If the connected-sum additivity assumption used in Corollaries 4.4 and 4.5 is proved from the variational principle, ribbonlength would become an almost additive knot energy, with each added summand contributing its own minimal value minus a fixed one-disk correction.
  • The method suggests a search template for further exact values: start from any union of stadium curves enclosing unit disks in a lattice that satisfies the separation bound, run the descent, and read off a formula; the rectangular family is one instance of this template.
  • Because the paper's minimality claims are restricted to standard diagram types, an independent search over all planar realisations of the Turk's head knot could determine whether $16+3\pi$ is a property of the knot itself or only of its standard diagram.
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

5 major / 4 minor

Summary. The paper studies flat ribbon knots in the plane and proposes a variational framework in a larger space of disk diagrams. It introduces a local criticality criterion (Lemma 4.1), then claims exact minimal ribbonlength values for several standard knot diagrams: the Salomon knot (8+2π), the Turk's head knot (16+3π), the granny and square knots (12+3π), connected sums of Salomon knots (8n+2π), and an infinite rectangular family (π(m+n)+8mn). A further family is claimed to be ribbonlength-critical with ribbonlength 8∑k+(n+1)π. The paper also states conjectures relating ribbonlength to crossing number and ropelength.

Significance. If the exact values are correct, they would be the first exact flat-ribbon ribbonlength values beyond the unknot and trefoil, and the disk-space/cs-curve framework would be a valuable tool for a notoriously difficult geometric knot theory problem. The paper gives explicit constructions and a concrete variational criterion, and it is honest about the distinction between criticality and global minimality in Theorem 5.3. However, the proofs as written contain several load-bearing gaps: connected-sum additivity is asserted rather than derived, a key lemma is internally inconsistent, and the claimed lower bounds in Theorems 4.2, 4.3, and 5.1 are not actually proved. The contribution is therefore potentially significant but not yet established at the level of rigor required for the claims.

major comments (5)
  1. [Corollaries 4.4 and 4.5] The connected-sum additivity rule is asserted but not proved. Corollary 4.5 states that in the connected sum of two trefoils 'one unit disk perimeter overlaps between the two trefoils' and subtracts π from the ribbonlength, but the shared boundary length is never computed from the disk-space variational principle of Section 3; it is not shown that a minimizer of the sum is the sum of minimizers or that the overlap is exactly one unit-disk perimeter. Corollary 4.4 has a factor-two ambiguity: if 'subtract π from each of the length components' refers to core lengths, the ribbonlength reduction is π, not 2π; if it refers to component ribbonlengths, the operation is not derived. Since the values 12+3π and 8n+2π are headline results, this missing additivity proof is load-bearing.
  2. [Lemma 4.1] The statement of Lemma 4.1(2) says the family of admissible perturbed cs curves is monotonically length-increasing, achieving a maximum of 5π/3 + 2, but the proof computes the limiting length as π/3 + 2 + π + π/3 + π/6 = 11π/6 + 2. These numbers differ, so the lemma as stated is internally inconsistent. Because Lemma 4.1 is later used as the criticality criterion for the infinite families in Section 5, this inconsistency must be resolved. Item (3) also asserts 'two saddle points' from the mountain pass lemma, but the hypotheses of the mountain pass lemma are not verified.
  3. [Theorem 4.3] The proof of minimality for the Turk's head diagram decomposes the diagram into two loops and then takes the ribbonlength to be half the sum of the two loop lengths. This ignores the geometry of the overlap or shared boundary between the two loops; no argument shows that the two lengths can simply be added. In addition, the claimed minimal length 16 + 2π for γ2, the 'perimeter of 9 disks with centres in the square lattice', is asserted without computation or reference; this value is not an immediate consequence of the stadium bound used elsewhere in the paper and needs a proof. The value 16+3π therefore rests on an unverified decomposition.
  4. [Theorem 5.1] The proof of minimality for the rectangular family consists of asserting that gradient descent converges to a lattice configuration that 'achieves all lower bounds, as each of the components are minimal perimeter stadium curves enclosing a finite number of disks.' This is not a lower-bound proof: a stadium curve is minimizing only for a collinear set of disks, and no argument is given that an arbitrary arrangement of m×n unit disks has core length at least 2π(m+n)+16mn. The convergence of the descent and preservation of the separation bound are also assumed rather than shown. Without these steps the exact value π(m+n)+8mn is not established.
  5. [Theorem 4.2] The proof of the Salomon value relies on the assertion that 'each of the disks D1, D2, D3, D4 intersects D0 at a single point' and that the final configuration 'attains all lower bounds, since the minimal perimeter enclosing 3 disks is given by a stadium curve of length 8+2π.' The latter statement is confusing (the diagram has five disks D0,...,D4) and no global lower bound for the Salomon diagram is derived from it. The local perturbation argument only rules out one type of obstruction, so the global minimality claim is not fully supported.
minor comments (4)
  1. [Lemma 4.1 and Section 2] The notation 'scs' appears in Lemma 4.1 and its proof but is never defined; Section 2 defines 'cs' curves, and the relation between the two should be clarified.
  2. [Figure 7 caption] The caption of Figure 7 refers to the 'doormat family', while the text calls it the 'Salomon family'; this terminology should be made consistent.
  3. [Corollary 4.4] The phrase 'subtract π from each of the length components' is ambiguous: it should specify whether the subtraction applies to core lengths or to ribbonlengths.
  4. [Section 2, Observation] The observation that the paper fixes standard diagrams rather than searching over all realizations of a knot is important and should appear in the introduction, because the exact values are not claimed for all diagrams of the knot.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found; the connected-sum additivity rule is a soundness gap, not a circular reduction.

full rationale

I walked the claimed derivation chain: Theorem 4.2 restricts to cs curves via the cited Theorem 2.4 from [1], then proves minimality by a disk-contact argument and the known perimeter of a stadium curve around three unit disks; this is a legitimate use of a prior theorem and an external geometric fact, not a restatement of the conclusion. Theorem 4.3 similarly uses Theorem 2.4 and decomposes the Turk's head diagram into two loops whose lengths are added; the additivity of the decomposition is a geometric claim about the diagram, not a circular reduction. Corollaries 4.4 and 4.5 do rely on the asserted overlap rules 'one unit disk perimeter overlaps between the two trefoils' and 'we must subtract a total of 2×2π', and Corollary 4.4 appears to mix core-length and ribbonlength reductions; but an unproved or even internally inconsistent additivity assumption is a correctness/soundness problem, not a case where the prediction is equivalent to its input by construction. The only self-citations are to prior theorems in [1] (existence of minimizers, cs regularity, Rib(trefoil)=6+2π); those are cited as established results with independent derivations and are not fitted parameters renamed as predictions. No fitted input is called a prediction, no uniqueness theorem is imported from the authors, and no known result is renamed. Hence there is no significant circularity.

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

No numerical parameters are fitted to data; the constants π and the integer n are exact geometric inputs. The main quantities, such as 8 + 2π, are derived rather than fitted. The 'disk space' and 'cs curves' are formalizations from [1], not new entities postulating unexplained phenomena.

assumptions (5)
  • standard math Length minimizers in each component of disk space exist and are cs curves (Theorem 2.4 of [1]).
    The proofs in Section 4 begin by reducing to cs representatives using this theorem from the author's prior paper; no proof is repeated here.
  • domain assumption The ribbon width convention sets Width = 2, so Rib(γ) = Length(γ)/2 (Definition 3.1).
    All final numbers depend on this normalization; a different width convention would change every stated value.
  • ad hoc to paper Connected sum additivity: the minimal ribbonlength of a connected sum of diagrams is the sum of the components' minimal ribbonlengths minus π per shared disk.
    Used to compute granny, square, and connected sums of Salomon knots in Corollaries 4.4 and 4.5; no full proof is given that global rearrangements cannot reduce the total further.
  • domain assumption The lower bound for disk diagrams is obtained by decomposing the diagram into minimal perimeter stadium curves enclosing disks.
    Theorem 4.2 and Theorem 5.1 rely on matching 'all lower bounds' from such stadium curves; for large numbers of disks the minimal perimeter is not known (Schürmann [11]), which the paper concedes.
  • ad hoc to paper The geometric gradient descent from a suitable initial cs diagram converges to a configuration that can be verified length-minimal or critical.
    Theorems 4.3 and 5.1 assert convergence and minimality without a convergence proof; the argument is diagram-dependent and supported by figures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Profiles of Critical Flat Ribbon Knots." pith.science (2026). https://pith.science/paper/26RUYILG

@misc{pith2026250604403,
  author       = {Pith},
  title        = {Pith review of: Profiles of Critical Flat Ribbon Knots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26RUYILG}},
  note         = {Machine review of arXiv:2506.04403}
}
read the original abstract

The main open problem in geometric knot theory is to provide a tabulation of knots based on an energy criterion, with the goal of presenting this tabulation in terms of global energy minimisers within isotopy classes, often referred to as ideal knots. Recently, the first examples of minimal length diagrams and their corresponding length values have been determined by Ayala, Kirszenblat, and Rubinstein. This article is motivated by the scarcity of examples despite several decades of intense research. Here, we compute the minimal ribbonlength for some well-known knot diagrams, including the Salomon knot and the Turk's head knot. We also determine the minimal ribbonlength for the granny knot and square knot using a direct method. We conclude by providing the ribbonlength for infinite classes of critical ribbon knots, along with conjectures aimed at relating ribbonlength to knot invariants in pursuit of a metric classification of knots.

Figures

Figures reproduced from arXiv: 2506.04403 by the authors.

Figure 1
Figure 1. Above: The Salomon knot has a minimal ribbonlength of 2π + 8. The dashed trace represents the core of the ribbon. Below: The second element of an infinite family of diagrams described in Section 5. The diagrams considered here are related to several planar representations of knots, which have been used across various interconnected contexts. For instance, Lomonaco and Kauffman presented a system of tiles to define q… view at source ↗
Figure 2
Figure 2. We start with an scs elastic-like curve and roll C over A clockwise to achieve the limit curve on the right. The left figure illustrates that the first variation vanishes since u · v = 0. 3π/2 to 3π/2 − π/3 = 4π/3, and therefore u2 · v is monotonically increasing during the rolling since v remains fixed, at the center of C, see [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Examples of Salomon knot disk diagrams. Left: there are two scenar￾ios under the descent. The disk D4 is free to approach D0, while disk D3 blocks D1. Right: a minimal-length Salomon disk diagram. This minimiser satisfies the separation bound. Theorem 4.2. The minimal ribbonlength for the standard Salomon knot diagram is 8 + 2π. Proof. Because length minimisers in disk space are cs curves (Theorem 2.4), we can consi… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Theorem 4.3. The minimal ribbonlength for the standard Turk’s head diagram is 16 + 3π. Proof. Since length minimisers in disk space are cs curves (see Theorem 2.4), we consider a suitable cs representative for the Turk’s head diagram to facilitate the application of th…
Figure 5
Figure 5. Figure 5: Left: Minimal ribbon knot diagrams for the granny knot. Right: Minimal ribbon knot diagrams for the square knot. Note there is a four-parameter family of these objects. We can perturb the top diagrams to obtain the bottom ones and vice versa without changing their leng…
Figure 6
Figure 6. Figure 6: Left: A rectangular disk diagram in a 2×2 arrangement, also known as the Salomon quadruple. Centre: A tightened disk diagram for the 2 × 2 arrangement. Right: A minimal ribbonlength 2 × 2 arrangement. unknotted loop (see [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Left: A diagram of the third element in the doormat family. Centre: A tightened disk diagram for the third element in the doormat family. Right: A critical ribbon diagram for the third element in the doormat family; minimality cannot be guaranteed due to [11], as for l…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 15 canonical work pages

  1. [1]

    Adams, A

    C. Adams, A. Kastner, A. Calderon, X. Jiang, G. Kehne, N. Mayer, and M. Smith, Volume and determinant densities of hyperbolic rational links, Journal of Knot Theory and Its Ramifications, vol. 24, no. 03, pp. 1550011, 2015

    work page 2015

  2. [2]

    Ayala, D

    J. Ayala, D. Kirszenblat, J.H. Rubinstein, Immersed flat ribbon knots, Israel Journal of Mathematics, to appear, https://arxiv.org/abs/2005.13168

    arXiv 2005

  3. [3]

    Cantarella, J.H.G

    J. Cantarella, J.H.G. Fu, R.B. Kusner, J.M. Sullivan, N.C. Wrinkle, Criticality for the Gehring link problem, Geometry & Topology, vol. 10, pp. 2055–2115, 2006

    work page 2006

  4. [4]

    Cantarella, R.B

    J. Cantarella, R.B. Kusner, J.M. Sullivan, On the minimum ropelength of knots and links, Inventiones Mathematicae, vol. 150, pp. 257-286, 2002

    work page 2002

  5. [5]

    Evans, V

    M.E. Evans, V. Robins, and S.T. Hyde, Ideal geometry of periodic entanglements, Proceedings of the Royal Society A, vol. 471, no. 2181, pp. 1-23, 2015

    work page 2015

  6. [6]

    Chaim Even-Zohar, Joel Hass, Nati Linial, Tahl Nowik, Universal knot diagrams, Journal of Knot Theory and Its Ramifications, vol. 28, no. 07, p. 1950031, 2019. doi: 10.1142/S0218216519500317

    doi:10.1142/s0218216519500317 2019

  7. [7]

    Folded ribbon knots in the plane

    E. Denne, Folded ribbon knots in the plane, arXiv:1807.00691v1

    work page Pith review arXiv

  8. [8]

    Y. Diao, C. Ernst, and E.J. Janse van Rensburg, Thickness of knots, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 126, pp. 293-310, 1999

    work page 1999

Show all 18 references
  1. [9]

    Gonzalez, J.H

    O. Gonzalez, J.H. Maddocks, Global curvature, thickness, and the ideal shapes of knots, Proceedings of the National Academy of Sciences (USA), vol. 96, pp. 4769-4773, 1999

    1999

  2. [10]

    Kauffman, Minimal flat knotted ribbons, in Series on Knots and Everything, Physical and Numerical Models in Knot Theory, World Scientific Publishing, 2005, pp

    L.H. Kauffman, Minimal flat knotted ribbons, in Series on Knots and Everything, Physical and Numerical Models in Knot Theory, World Scientific Publishing, 2005, pp. 495-506

    2005

  3. [11]

    A Gordian Pair of Links

    Kusner, R., Kusner, W. A Gordian Pair of Links. Geom Dedicata 217, 47 (2023). https://doi.org/10.1007/s10711-023-00783-1

    2023 doi

  4. [12]

    Litherland, J

    R.A. Litherland, J. Simon, O. Durumeric, E. Rawdon, Thickness of knots, Topology and its Applications, vol. 91, no. 3, pp. 233-244, 1999. doi: 10.1016/S0166-8641(97)00210-1

    1999 doi

  5. [13]

    Lomonaco and L

    S.J. Lomonaco and L. Kauffman, Quantum knots and mosaics, Quantum Information Processing, vol. 7, no. 2-3, pp. 85-115, 2008

    2008

  6. [14]

    Grosberg and S

    A. Grosberg and S. Nechaev, Algebraic invariants of knots and disordered Potts model, Journal of Physics A: Mathematical and General, vol. 25, no. 17, pp. 4659, 1992

    1992

  7. [15]

    Jung, Man & His Symbols, Bantam Doubleday Dell Publishing Group, 1997

    C.G. Jung, Man & His Symbols, Bantam Doubleday Dell Publishing Group, 1997

    1997

  8. [16]

    Nechaev, Statistics of knots and entangled random walks, World Scientific, 1996, https://arxiv.org/abs/cond-mat/9812205

    S.K. Nechaev, Statistics of knots and entangled random walks, World Scientific, 1996, https://arxiv.org/abs/cond-mat/9812205

    1996 arXiv

  9. [17]

    Sch \"u rmann, ``On extremal finite packings,'' Discrete & Computational Geometry, vol

    A. Sch \"u rmann, ``On extremal finite packings,'' Discrete & Computational Geometry, vol. 28, no. 3, pp. 389--403, 2002

    2002

  10. [18]

    L. Younes. Shapes and Diffeomorphisms. Springer, 2010

    2010

Pith tools

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