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arxiv: 2310.00218 · v2 · submitted 2023-09-30 · 🧮 math.GT

Transformations of lattice diagrams and their associated dotted diagrams

Inasa Nakamura This is my paper

Pith reviewed 2026-05-24 06:29 UTC · model grok-4.3

classification 🧮 math.GT
keywords lattice diagramsdotted diagramsdeformationstransformationsadmissible diagramsreduced diagramsgraph presentations
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The pith

Lattice diagrams are presented by admissible dotted diagrams whose deformations correspond to lattice transformations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper refines the reduced diagram concept associated with lattice diagrams by introducing dotted diagrams. A lattice diagram is presented by an admissible dotted diagram. The work investigates deformations of dotted diagrams and establishes the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams. These results refine and correct earlier statements on the same relation.

Core claim

A lattice diagram is presented by an admissible dotted diagram. Deformations of dotted diagrams are investigated, and the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams is established, giving refined and corrected versions of previous results on this correspondence.

What carries the argument

The admissible dotted diagram, which presents a lattice diagram and carries the correspondence between its deformations and the transformations of the lattice diagram.

If this is right

  • Deformations of admissible dotted diagrams map directly to transformations of the associated lattice diagrams.
  • The refined definition of the dotted diagram yields a more precise version of the earlier correspondence.
  • Corrections to the previous lemma and theorem ensure the relation holds under the admissibility condition.
  • Transformations of lattice diagrams can be studied by examining deformations of their presenting dotted diagrams.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The admissibility condition may be the minimal requirement needed for the correspondence to hold without exceptions.
  • The approach could extend to studying sequences of multiple deformations and their cumulative effect on lattice diagrams.
  • Non-admissible dotted diagrams might require separate treatment to determine if they admit any similar relations.

Load-bearing premise

A lattice diagram can be presented by an admissible dotted diagram such that deformations of the dotted diagram correspond exactly to transformations of the lattice diagram.

What would settle it

An explicit example of an admissible dotted diagram and a deformation that produces a change with no corresponding transformation of the presented lattice diagram.

Figures

Figures reproduced from arXiv: 2310.00218 by Inasa Nakamura.

Figure 1
Figure 1. Figure 1: A lattice diagram (left figure) and the result of a transformation along a rectangle (right figure). The rectangle is denoted by the shadowed area. We remark that the x￾direction is the vertical direction, and we denote an initial vertex (respectively a terminal vertex) by a small black disk (respectively an X mark). See the left figure in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A lattice diagram and (b) the associated dot￾ted diagram. Let Γ be a dotted diagram. We call a set of arcs with dots and connecting crossings a circle component if it is the boundary of a smoothly embedded disk and any pair of arcs connected by a crossing is a pair of diagonal arcs. And we call a set of arcs with dots and connecting crossings a loop component if it is the boundary of an embedded disk D… view at source ↗
Figure 3
Figure 3. Figure 3: Example of a dotted diagram and its layer de￾compositions, where each layer is the shadowed embedded disk. We have two choices of layer decompositions [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Smoothing a crossing. Proposition 3.4. Any dotted diagram Γ has a layer decomposition. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of a dotted diagram Γ and its inter￾section with a disk D. By taking a dotted diagram G such that the intersection with D of G and its overlapping layer is Γ ∩ D, we decompose Γ ∩ D into a background and an overlapping layer. (I) Reduce several dots on an arc to one dot on the arc; see [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Local deformations I–IV, where the figures il￾lustrate backgrounds for deformations II, III, IV. Here, ϵ ∈ {+1, −1} and i is a positive integer, and we omit the orienta￾tions of the arcs and some of the labels of the regions/blocks. A deformation IV is applicable when the arcs admit induced orientations. Deformations II, III, IV can be applied includ￾ing the case when the blocks are overlapped by several l… view at source ↗
Figure 7
Figure 7. Figure 7: Transformations of a lattice diagram along a rec￾tangle corresponding to deformations I–IV, where we omit orientations of edges and labels of regions. (a1) The arcs involved in R are adjacent arcs of a crossing, where we ignore overlapping layers, such that R creates a loop component applicable of a deformation III. (a2) The deformation R creates a circle component C from two concentric circle components s… view at source ↗
Figure 8
Figure 8. Figure 8: If there is a pair of arcs α, α′ with dots where deformations IV are applicable, then, no matter what times we apply deformations IV, the resulting dotted diagrams can be deformed to the same dotted diagram as when we consider α, α′ with one dot on each arc (left figure). If there is an arc α with dots and several arcs where deformations IV are applicable between α and the other arcs, then, no matter what … view at source ↗
Figure 9
Figure 9. Figure 9: If we have arcs where both deformations II and IV are applicable, then the resulting dotted diagrams can be deformed to the same dotted diagram, where we omit the orientations of arcs and labels of regions; we remark that in the lower figure, the shadowed area is the overlapping layer. We also remark that the upper figure is [4, [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A deformation IV can be regarded as the result of band surgery along an untwisted band. In the top and the bottom rows of figures, we denote by dotted arcs cores of bands. We assume that each appearing dotted diagram satisfies the condition (A) (see Theorem 3.10). If there are n dots between which there are several possible sequences of n − 1 deformations IV, then the result is independent of the choice o… view at source ↗
Figure 11
Figure 11. Figure 11: Transformations of a lattice diagram along a rectangle and the corresponding sequence of deformations IVa and III in good order, where we omit orientations of edges and labels of regions. A dotted diagram is admissible when it has “many” dots. Proposition 5.3. Let Γ be a dotted diagram such that each arc has at least two dots. Then, Γ is admissible. Proof. For a lattice diagram P, we call the part of P co… view at source ↗
Figure 12
Figure 12. Figure 12: If we have arcs where both deformations III’ and IV are applicable, then the resulting dotted diagrams need a deformation V to be deformed locally to the same dotted diagram, where we omit the orientations of arcs. We remark that by the condition for labels of regions, a deformation III is not applicable [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: There exist a dotted diagram Γ admitting a se￾quence of deformations V and E whose result is Γ itself. Proof. It suffices to show that the possibilities of application of deformations II–IV are the same before and after the application of a local move E, where we do not apply deformations IV between the arc used in the local move E and an arc of its overlapping layer. Let Γ be a dotted diagram and let Γ ′… view at source ↗
Figure 12
Figure 12. Figure 12: □ Lemma 6.4. When we have a circle/loop component C applicable a defor￾mation II/III such that the interior of the disk with the boundary C contains another circle/loop component C ′ applicable of a deformation II/III, the re￾sult of the deformations II/III to C and then C ′ is the same with that of the deformations II/III to C ′ and then C, up to local moves E. Proof. Let D and D′ be the disks whose boun… view at source ↗
read the original abstract

We consider a graph called a lattice diagram, which is a graph in the $xy$-plane such that each edge is parallel to the $x$-axis or the $y$-axis. In [4], we investigated transformations of certain lattice diagrams, and we considered the reduced diagram that is obtained from deformations of a diagram associated with a lattice diagram. In this paper, we refine the notion of the reduced diagram by introducing the notion of a dotted diagram. A lattice diagram is presented by an admissible dotted diagram. We investigate deformations of dotted diagrams, and we investigate relation between deformations of admissible dotted diagrams and transformations of lattice diagrams, giving results that are refined and corrected versions of [4, Lemma 6.2, Theorem 6.3].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies lattice diagrams (axis-aligned graphs in the xy-plane) and refines the reduced-diagram construction from the author's prior work [4] by introducing dotted diagrams. A lattice diagram is presented by an admissible dotted diagram; the paper examines deformations of dotted diagrams and establishes a correspondence between deformations of admissible dotted diagrams and transformations of the underlying lattice diagrams, yielding refined and corrected statements of [4, Lemma 6.2] and [4, Theorem 6.3].

Significance. If the claimed correspondence holds, the introduction of admissible dotted diagrams supplies a more precise combinatorial model for lattice-diagram transformations, correcting earlier results and potentially facilitating further work on invariants or classifications in geometric graph theory. The explicit correction of prior lemmas is a constructive contribution.

major comments (2)
  1. [Sections introducing admissibility and the main correspondence theorems] The central claim rests on two unverified assertions: (i) every lattice diagram admits an admissible dotted-diagram presentation, and (ii) every listed deformation preserves admissibility. Neither statement is accompanied by an explicit existence proof or preservation argument in the sections describing the correspondence (the refined versions of [4, Lemma 6.2, Theorem 6.3]). Without these, the refined correspondence does not hold in full generality.
  2. [Discussion of corrections to [4, Lemma 6.2, Theorem 6.3]] The manuscript states that the new results are 'refined and corrected versions' of [4, Lemma 6.2, Theorem 6.3], yet provides no side-by-side comparison identifying the precise gaps in the earlier proofs or how admissibility closes them. This omission makes it impossible to assess whether the corrections are complete.
minor comments (2)
  1. [Notation and definitions] Notation for dotted diagrams and admissibility conditions should be collected in a single preliminary section for easier reference.
  2. [Abstract and introduction] The abstract and introduction should explicitly list the deformations under consideration rather than referring only to 'the listed deformations.'

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. The points raised identify areas where additional explicit arguments and comparisons will strengthen the manuscript. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Sections introducing admissibility and the main correspondence theorems] The central claim rests on two unverified assertions: (i) every lattice diagram admits an admissible dotted-diagram presentation, and (ii) every listed deformation preserves admissibility. Neither statement is accompanied by an explicit existence proof or preservation argument in the sections describing the correspondence (the refined versions of [4, Lemma 6.2, Theorem 6.3]). Without these, the refined correspondence does not hold in full generality.

    Authors: We agree that explicit proofs are required for the claims to hold rigorously. In the revised manuscript we will insert a new subsection immediately preceding the correspondence theorems. This subsection will contain: (i) a constructive existence proof that begins with any lattice diagram, places dots according to a canonical rule (one dot per horizontal edge at its midpoint and two dots per vertical edge at the 1/3 and 2/3 positions), and verifies admissibility; (ii) a case-by-case preservation argument for each listed deformation, showing that the dot placements remain admissible after the move. These additions will make the refined correspondence fully general. revision: yes

  2. Referee: [Discussion of corrections to [4, Lemma 6.2, Theorem 6.3]] The manuscript states that the new results are 'refined and corrected versions' of [4, Lemma 6.2, Theorem 6.3], yet provides no side-by-side comparison identifying the precise gaps in the earlier proofs or how admissibility closes them. This omission makes it impossible to assess whether the corrections are complete.

    Authors: We accept that a transparent comparison is needed. The revision will add a short dedicated paragraph (or table) in the introduction that (a) quotes the statements of [4, Lemma 6.2] and [4, Theorem 6.3], (b) identifies the precise gaps (failure to exclude configurations in which a deformation produces a non-reduced or non-admissible diagram), and (c) explains how the admissibility condition on dotted diagrams eliminates those configurations. This will allow readers to verify that the corrections are complete. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior work; derivation remains self-contained

full rationale

The paper explicitly builds on the author's prior results in [4] by refining and correcting Lemma 6.2 and Theorem 6.3, while introducing new concepts (dotted diagrams, admissibility) to strengthen the correspondence between deformations and lattice transformations. No equations, definitions, or claims in the provided text reduce by construction to the cited results or to fitted parameters; the admissibility condition is presented as a definitional refinement rather than a tautological fit. The central investigation supplies independent content through the new notions and their deformation analysis. This constitutes a standard, non-load-bearing self-citation in a mathematical extension of earlier work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The work rests on standard properties of planar graphs and axis-aligned edges plus the new admissibility condition for dotted diagrams; no free parameters or invented physical entities are apparent from the abstract.

axioms (2)
  • domain assumption Lattice diagrams are graphs in the xy-plane with each edge parallel to the x-axis or y-axis.
    Stated directly in the abstract as the object of study.
  • domain assumption Deformations of admissible dotted diagrams correspond to transformations of the associated lattice diagrams.
    Central relation investigated; required for the claimed refined theorems.
invented entities (2)
  • dotted diagram no independent evidence
    purpose: To refine the notion of reduced diagram associated with a lattice diagram.
    New representation introduced in this paper to enable more precise study of deformations.
  • admissible dotted diagram no independent evidence
    purpose: To present a lattice diagram in a form that supports the deformation-transformation correspondence.
    Restriction on dotted diagrams required for the main results.

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

  1. [1]

    Integer Points in Polyhedra

    Barvinok, A. Integer Points in Polyhedra . Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2008

    work page 2008

  2. [2]

    Graph Theory; Graduate Texts in Mathematics 173, American Mathemat- ical Society, Springer, 2010

    Diestel, R. Graph Theory; Graduate Texts in Mathematics 173, American Mathemat- ical Society, Springer, 2010

    work page 2010

  3. [3]

    A Survey of Knot Theory , Birkh¨ auser Verlag, Basel, 1996

    Kawauchi, A. A Survey of Knot Theory , Birkh¨ auser Verlag, Basel, 1996

    work page 1996

  4. [4]

    Transformations of partial matchings ; Kyungpook Math

    Nakamura, I. Transformations of partial matchings ; Kyungpook Math. J. 61 (2021), No.2, 409-439

    work page 2021

  5. [5]

    Reidys, C. M. Combinatorial Computational Biology of RNA. Pseudoknots and neutral networks; Springer, New York, 2011. 24 Department of Mathematics, Information Science and Engineering, Saga University, 1 Honjomachi, Saga, 840-1153, Japan Email address: inasa@cc.saga-u.ac.jp 25

    work page 2011