Compact surfaces as configuration spaces of mechanical linkages (Q5935807)

The main result is easy to describe: Mechanical linkages, i.e., mechanisms in the Euclidean plane \(E^2\) built up exclusively from rigid bars joined along flexible links, are considered and their configuration spaces are investigated, i.e., the totality of all their admissible positions in the Euclidean plane. The authors prove that for any compact orientable closed surface there exists a mechanical linkage having a configuration space of the same type. The proof is constructive and not simple. Indications are given that the result may be extended to the non-orientable case, but no proof is presented for this.