Algebraic & Geometric Topology

Topology of random linkages

Michael Farber

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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.

Article information

Algebr. Geom. Topol., Volume 8, Number 1 (2008), 155-171.

Received: 21 June 2007
Accepted: 18 December 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R80: Discriminantal varieties, configuration spaces 55N99: None of the above, but in this section
Secondary: 55M99: None of the above, but in this section

linkage polygon space random manifold betti number


Farber, Michael. Topology of random linkages. Algebr. Geom. Topol. 8 (2008), no. 1, 155--171. doi:10.2140/agt.2008.8.155.

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