The genus of configuration curves of planar linkages is generically odd

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.