The genus of configuration curves of planar linkages is generically odd

TL;DR

This paper proves that the genus of the algebraic curve associated with a generic one-degree-of-freedom planar linkage is always odd or zero, using tropical geometry techniques.

Contribution

It establishes that the genus of configuration curves for such linkages is always odd or zero, revealing a fundamental topological property.

Findings

Genus is always odd for non-zero cases

Genus can be zero for special configurations

Proof utilizes tropical geometry methods

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.