Compact Surfaces as Configuration Spaces of Mechanical Linkages

There exists a homeomorphism between any compact orientable closed surface and the configuration space of an appropriate mechanical linkage defined by a weighted graph embedded in the Euclidean plane. 1. I n t r o d u c t i o n A m e c h a n i c a l l inkage G is a mechanism in the Euclidean plane R 2 that is built up exclusively from rigid bars joined along flexible links. Some links of the linkage may be pinned down with respect to a fixed frame of reference. The conf igur a t i o n s p a c e [G] of a mechanical linkage G is the totality of all its admissible positions in the Euclidean plane. In section 4 we present the constructive proof of the main result: Received October 14, 1998 and in revised form October 20, 1999