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On Galois groups of type-1 minimally rigid graphs

Mehdi Makhul, Josef Schicho, Audie Warren

Abstract

For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. Here we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.

On Galois groups of type-1 minimally rigid graphs

Abstract

For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. Here we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.
Paper Structure (11 sections, 7 theorems, 57 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Galois group of a minimally rigid graph
  4. Signed area and the Cayley-Menger determinant
  5. Type-1 graphs
  6. Proof of Theorem \ref{['thm: main']}
  7. Proof that $\psi(\operatorname{Gal}(\mathcal{G})) = \psi(J)$
  8. Proof that $|\operatorname{Gal}(\mathcal{G}) \cap \ker(\psi)| =|J \cap \ker(\psi)|$
  9. Proof that $|J \cap \ker(\psi)|=2^k$
  10. Proof that $|\operatorname{Gal}(\mathcal{G}) \cap \ker(\psi)| = 2^k$
  11. A simple example applying Theorem \ref{['thm: main']}

Key Result

Theorem 2.1

By the following two rules we can construct all minimally rigid graphs, starting from a single edge.

Figures (1)

  • Figure 1: The graph $\mathcal{G}$

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • ...and 2 more