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Transformations of lattice diagrams and their associated dotted diagrams

Inasa Nakamura

TL;DR

The paper refines the study of lattice diagrams by introducing dotted diagrams and a layered deformation framework. It proves that the reduced diagram of a dotted diagram is unique up to controlled local moves and that deformation sequences are finite, yielding a corrected and strengthened link between dotted-diagram deformations and lattice dissolutions. Central results connect good, ordered deformations to minimal-area dissolutions of the associated lattice diagrams, enabling criteria to determine when a diagram can dissolve to empty with minimal area. These contributions clarify the correspondence between combinatorial diagram deformations and geometric transformations, with implications for understanding transformations and optimization in lattice-based diagrammatic models.

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].

Transformations of lattice diagrams and their associated dotted diagrams

TL;DR

The paper refines the study of lattice diagrams by introducing dotted diagrams and a layered deformation framework. It proves that the reduced diagram of a dotted diagram is unique up to controlled local moves and that deformation sequences are finite, yielding a corrected and strengthened link between dotted-diagram deformations and lattice dissolutions. Central results connect good, ordered deformations to minimal-area dissolutions of the associated lattice diagrams, enabling criteria to determine when a diagram can dissolve to empty with minimal area. These contributions clarify the correspondence between combinatorial diagram deformations and geometric transformations, with implications for understanding transformations and optimization in lattice-based diagrammatic models.

Abstract

We consider a graph called a lattice diagram, which is a graph in the -plane such that each edge is parallel to the -axis or the -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].
Paper Structure (12 sections, 16 theorems, 5 equations, 13 figures, 1 table)

This paper contains 12 sections, 16 theorems, 5 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Transformations of lattice diagrams and the areas of transformations
  3. Transformations of lattice diagrams
  4. Areas of transformations
  5. Dotted diagrams and their reduced diagrams
  6. Dotted diagrams
  7. Layer decomposition of a dotted diagram
  8. Background and overlapping layers of $\Gamma \cap D$
  9. Cores of deformations IV
  10. Proofs of Theorems \ref{['prop3-5']} and \ref{['r-prop3-5']}
  11. Deformations of admissible dotted diagrams and dissolutions of lattice diagrams
  12. Remark and Lemmas

Key Result

Theorem 2.3

Let $P$ be a lattice diagram. We consider a dissolution of $P$, $\mathrm{Ver}_0(P)=\Delta_0 \to \Delta_1\to \cdots \to \Delta_k=\mathrm{Ver}_1(P)$, such that each $\Delta_{j-1} \to \Delta_j$ is a transformation by a rectangle $R_j$ ($j=1,2,\ldots, k)$. We regard each $R_j$ as a lattice diagram, whos and Further, when $P$ satisfies either the condition $(1)$ or $(2)$ in N, there exist dissolutions

Figures (13)

  • 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).
  • Figure 2: (a) A lattice diagram and (b) the associated dotted diagram.
  • Figure 3: Example of a dotted diagram and its layer decompositions, where each layer is the shadowed embedded disk. We have two choices of layer decompositions.
  • Figure 4: Smoothing a crossing.
  • Figure 5: Example of a dotted diagram $\Gamma$ and its intersection with a disk $D$. By taking a dotted diagram $G$ such that the intersection with $D$ of $G$ and its overlapping layer is $\Gamma \cap D$, we decompose $\Gamma \cap D$ into a background and an overlapping layer.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 3.1
  • Definition 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • Example 3.5
  • ...and 32 more