A planar linkage is a collection of rigid bars joined at pivot points, constrained to move in the plane. Remove one bar from a minimally rigid framework, and the resulting mechanism has one degree of freedom — it can move along a one-parameter family of configurations. The set of these configurations, modulo rigid motions, forms an algebraic curve. The genus of this curve — a topological invariant measuring how many “holes” the configuration space has — depends only on the graph structure of the linkage, not on the specific edge lengths.
Schicho, Tewari, and Warren (arXiv:2603.11641) prove that this genus is always odd, unless it is zero. A linkage with a genus-2 configuration space does not exist. Nor genus 4, nor 6, nor any positive even number. The configuration curve of a planar one-degree-of-freedom mechanism is always a sphere (genus 0) or has an odd number of handles.
The proof uses tropical geometry — a combinatorial shadow of algebraic geometry where curves become graphs and intersection theory becomes counting. The tropical approach allows the authors to compute genera from the graph structure directly, and the parity constraint emerges from the combinatorics of how rigid subgraphs intersect when one bar is removed.
The structural point is that mechanical motion spaces have a hidden parity constraint. You can build linkages with arbitrarily complicated configuration curves — high genus, many branches, intricate topology. But the complexity is always odd. This is not a design choice. It is a consequence of the algebraic structure that rigid frameworks impose on their configuration spaces. The mechanism cannot explore a configuration space with an even number of handles, not because of any physical limitation, but because the equations that define bar-and-joint rigidity in the plane algebraically forbid it.