Topology of random linkages

Michael Farber*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 R3. We also prove results about higher moments of Betti numbers.

Original languageEnglish
Pages (from-to)155-171
Number of pages17
JournalAlgebraic and Geometric Topology
Issue number1
StatePublished - 2008
Externally publishedYes


  • Betti number
  • Linkage
  • Polygon space
  • Random manifold


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