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Flexible placements of periodic graphs in the plane

Sean Dewar

TL;DR

NBAC-colourings are introduced for the corresponding quotient gain graphs to identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed.

Abstract

Given a periodic graph, we wish to determine via combinatorial methods whether it has periodic embeddings in the plane that -- via motions that preserve edge-lengths and periodicity -- can be continuously deformed into another non-congruent embedding of the graph. By introducing NBAC-colourings for the corresponding quotient gain graphs, we identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed. We further characterise with NBAC-colourings which 1-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity, and characterise in special cases which 2-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity.

Flexible placements of periodic graphs in the plane

TL;DR

NBAC-colourings are introduced for the corresponding quotient gain graphs to identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed.

Abstract

Given a periodic graph, we wish to determine via combinatorial methods whether it has periodic embeddings in the plane that -- via motions that preserve edge-lengths and periodicity -- can be continuously deformed into another non-congruent embedding of the graph. By introducing NBAC-colourings for the corresponding quotient gain graphs, we identify which periodic graphs have flexible embeddings in the plane when the lattice of periodicity is fixed. We further characterise with NBAC-colourings which 1-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity, and characterise in special cases which 2-periodic graphs have flexible embeddings in the plane with a flexible lattice of periodicity.

Paper Structure

This paper contains 25 sections, 50 theorems, 112 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function fields and valuations
  4. Gain graphs
  5. Rigidity and flexibility for k-periodic frameworks
  6. NBAC-colourings and flexibility in the plane
  7. NBAC-colourings
  8. k-periodic frameworks in the plane
  9. Active NBAC-colourings
  10. Key tools
  11. Characterising fixed lattice flexible frameworks
  12. Necessary conditions for fixed lattice flexibility
  13. Constructing fixed lattice flexible frameworks
  14. Characterising flexible 1-periodic frameworks
  15. Necessary conditions for 1-periodic flexibility
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\mathcal{C}$ be an algebraic curve in $\mathbb{C}[X_1, \ldots,X_n]$ and let $f \in \mathbb{C}[x_1, \ldots,x_n]$. Then one of the following holds:

Figures (13)

  • Figure 1: (Left): A rigid placement of $K_2 \times K_3$ in the plane. As $K_2 \times K_3$ is a Laman graph, almost all placements will give a rigid framework. (Right): A flexible placement of the same graph.
  • Figure 2: (Left): A framework $(\mathcal{G},\mathcal{P})$ with $2$-periodic symmetry. (Right): A corresponding triple $(G,p,L)$ with $L := 2 I_2$, where $I_2$ is the $2 \times 2$ identity matrix.
  • Figure 3: A $\Gamma$-gain graph with $a,b,c,d \in \Gamma$. We represent any edge $(v,w,\gamma)$ by an arrow from $v$ to $w$ with a label $\gamma$, and we represent any edge $(v,w,0)$ by an undirected and unlabelled edge from $v$ to $w$.
  • Figure 4: A switching operation at $u$ by $\mu$.
  • Figure 5: A surjective colouring $\delta$ of a $\Gamma$-gain graph. If $\alpha \notin \langle \beta \rangle$, $\beta \notin \langle \alpha -\gamma \rangle$ and $\gamma \neq 0$, then $\delta$ is a NBAC-colouring.
  • ...and 8 more figures

Theorems & Definitions (125)

  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.1
  • ...and 115 more