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The genus of configuration curves of planar linkages is generically odd

Josef Schicho, Ayush Kumar Tewari, Audie Warren

Abstract

A one-degree-of-freedom graph is a graph obtained from a minimally rigid graph in the plane and removing an edge. For such graph, the set of realisations with fixed edge length, modulo rotations and reflections, is an algebraic curve. The genus of a connected component for generic edge lengths is a number that depends only on the graph. We prove that this genus is always odd, unless it is zero. The proof is based on tropical geometry.

The genus of configuration curves of planar linkages is generically odd

Abstract

A one-degree-of-freedom graph is a graph obtained from a minimally rigid graph in the plane and removing an edge. For such graph, the set of realisations with fixed edge length, modulo rotations and reflections, is an algebraic curve. The genus of a connected component for generic edge lengths is a number that depends only on the graph. We prove that this genus is always odd, unless it is zero. The proof is based on tropical geometry.
Paper Structure (12 sections, 8 theorems, 33 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard definitions
  4. The realisation space
  5. The subgraph map
  6. Tropicalising the fibers of $S_G$
  7. Fibers of $S_G$
  8. Two specific fibers of $S_G$
  9. The tropical varieties $\overline{X}$ and $\overline{Y}$
  10. Irreducibility of $F_1$
  11. Applying the Riemann-Hurwitz Theorem
  12. Further remarks

Key Result

Theorem 1

Let $G$ be a one degree of freedom graph, and let $C$ be one component of the generic configuration curve of $G$. Then either $g(C)$ is odd, or $g(C) =0$, in which case $G$ is given by two minimally rigid subgraphs $G_1$ and $G_2$ which have one vertex in common.

Figures (1)

  • Figure 1: Top left: A graph $G$ of genus five. Top right: The tropicalisation of its generic fiber. Bottom left: A projected image of the algebraic generic fiber $F_1$ (two real components). Bottom right: A projected image of the algebraic special fiber $F_2$ (two real components), with a visible symmetry. (The tropicalisation is also symmetric, but the symmetry is not visible in the projection.)

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:equalandsmooth']}
  • Lemma 3
  • Lemma 4
  • proof : Proof of \ref{['lem:connectedmeansirreducible']}
  • ...and 5 more