Betti numbers of configuration spaces of mechanical linkages (known also as polygon
spaces) depend on a large number of parameters – the lengths of the bars of the
linkage. Motivated by applications in topological robotics, statistical shape
theory and molecular biology, we view these lengths as random variables and
study asymptotic values of the average Betti numbers as the number of links
tends to infinity. We establish a surprising fact that for a reasonably ample
class of sequences of probability measures the asymptotic values of the
average Betti numbers are independent of the choice of the measure. The
main results of the paper apply to planar linkages as well as for linkages in
. We
also prove results about higher moments of Betti numbers.
Keywords
linkage, polygon space, random manifold, betti number
