Topology of randon linkages

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

💡 Research Summary

The paper investigates the topology of configuration spaces of mechanical linkages—also known as polygon spaces—when the bar lengths are treated as random variables. For a linkage consisting of n bars, the set of all possible closed configurations (up to rigid motions) forms a manifold M(ℓ) whose Betti numbers β_k(M(ℓ)) encode the number of k‑dimensional holes. Classical studies fix a length vector ℓ and compute β_k for particular families, but in many applications (topological robotics, statistical shape analysis, molecular biology) the lengths are uncertain and naturally modeled by probability distributions. The authors therefore consider the average Betti numbers
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