The Euler characteristic of the configuration space of planar spidery linkages

The topology of configuration spaces of planar mechanical linkages has been studied by many people. Among these linkages, polygonal and spidery ones are particularly important. We call the configuration space of a polygonal linkage a polygon space. The study of polygon spaces was originated by Hausmann [7], Kapovich and Millson [10] and Walker [15]. Their study has a long history and has begotten many references. For example, the homology groups of polygon spaces of arbitrary edge lengths have been determined by Farber and Schutz [6].