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Generic infinitesimal rigidity for rotational groups in the plane

Alison La Porta, Bernd Schulze

Abstract

In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the centre of rotation. This extends the corresponding results for these groups in the free action case obtained by R. Ikeshita and S. Tanigawa in 2015, and our recent results for the reflection group and the rotational groups of order 2 and 3 in the non-free action case. The characterisations are given in terms of sparsity counts on the corresponding group-labelled quotient graphs, and are obtained via symmetry-adapted versions of recursive Henneberg-type graph constructions. For rotational groups of even order at least 8, we show that the sparsity counts alone are not sufficient for symmetry-generic infinitesimal rigidity.

Generic infinitesimal rigidity for rotational groups in the plane

Abstract

In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the centre of rotation. This extends the corresponding results for these groups in the free action case obtained by R. Ikeshita and S. Tanigawa in 2015, and our recent results for the reflection group and the rotational groups of order 2 and 3 in the non-free action case. The characterisations are given in terms of sparsity counts on the corresponding group-labelled quotient graphs, and are obtained via symmetry-adapted versions of recursive Henneberg-type graph constructions. For rotational groups of even order at least 8, we show that the sparsity counts alone are not sufficient for symmetry-generic infinitesimal rigidity.

Paper Structure

This paper contains 19 sections, 26 theorems, 41 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Infinitesimal rigidity of symmetric frameworks
  3. Gain graphs
  4. Balanced, near-balanced and S(k,j) gain graphs
  5. Balancedness
  6. Near-balancedness
  7. S(k,j) gain graphs
  8. Gain sparsity of a gain graph
  9. Gain graph extensions
  10. Blockers of a reduction
  11. The union of two blockers, one of which is a general-count blocker
  12. The union of two blockers, one of which is (2,3)-tight
  13. The union of two blockers with non-empty edge set
  14. A gain-tight graph admits a reduction
  15. Applying a 1-reduction at a vertex with exactly one neighbour
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $k\geq4$, and $(G,p)$ be a $\mathcal{C}_k$-symmetric framework. The spaces of trivial $\rho_0$-,$\rho_1$- and $\rho_{k-1}$-symmetric infinitesimal motions all have dimension 1. For $2\leq j\leq k-2$, the space of trivial $\rho_j$-symmetric infinitesimal motions has dimension 0.

Figures (7)

  • Figure 1: A $\Gamma$-symmetric graph and its $\Gamma$-gain graph. Here, $\Gamma\simeq\mathbb{Z}_6$ through an isomorphism which sends $\gamma\in\Gamma$ to 1. The unlabelled edges have gain $\text{id}$.
  • Figure 2: (a) is a proper near-balanced $\Gamma$-gain graph with $\Gamma$-lifting (b). (c) is a $S_0(9,j)$$\Gamma$-gain graph, where $|\Gamma|=9$, and (d) is its $\Gamma$-lifting. In (a,b), the unlabelled edges have gain $\text{id}$.
  • Figure 3: Examples of extensions. (a) is a 0-extension, where the gains $\alpha$ and $\beta$ are arbitrary. (b) is a loop-1-extension, where $\alpha\neq\textrm{id}$ and $\beta$ is a arbitrary. (c) is a 1-extension, where $\alpha=\beta\lambda^{-1}$ and $\delta$ is arbitrary. (d) is a 2-vertex-extension, where $\alpha$ and $\beta$ are arbitrary, and $\gamma$ is the generator of $\Gamma$ which corresponds to 1 in $\mathbb{Z}_k$. In (a,b,c), any one of the vertices incident to $v$ may be the fixed vertex.
  • Figure 4: Two instances of a vertex $v$ of degree 3. In both cases $v$ has two neighbours, one of which is fixed. In (a) there is an edge between the neighbours of $v$, in (b) there isn't.
  • Figure 5: Base graphs for $k$-fold rotation for $\rho_0,\rho_1$ and $\rho_{k-1}$. All edges may be labelled freely, with the only restriction that loops must have non-identity gains.
  • ...and 2 more figures

Theorems & Definitions (50)

  • Proposition 2.1
  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Example 3.5
  • Proposition 3.6
  • Definition 3.7
  • Definition 3.8
  • Remark 3.9
  • ...and 40 more