Volume 8, issue 1 (2008)

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ISSN (electronic): 1472-2739
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Topology of random linkages

Michael Farber

Algebraic & Geometric Topology 8 (2008) 155–171

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 n 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 3. We also prove results about higher moments of Betti numbers.

linkage, polygon space, random manifold, betti number
Mathematical Subject Classification 2000
Primary: 55R80, 55N99
Secondary: 55M99
Received: 21 June 2007
Accepted: 18 December 2007
Published: 8 February 2008
Michael Farber
Department of Mathematical Sciences
University of Durham