The configuration space of planar spidery linkages

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has n arms such that each arm is of length 1 + 1 and has a rotational joint in the middle, and that the endpoint of the kth arm is fixed to Re 2 ( k − 1 ) π n i . Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given.