Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.2997

Topology of randon linkages

Michael Farber
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.ATmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Farber. 2007-08-22. Topology of randon linkages. https://arxiv.org/abs/0708.2997

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Topological Hochschild homology of truncated Brown-Peterson spectra II

We compute topological Hochschild homology of $\mathbb{E}_3$-MU-algebra forms of the second truncated Brown-Peterson spectrum with Adams summand coefficients at $p=2$ and conditionally at arbitrary primes. We also provide a new computational tool, a variant of the Brun spectral sequence, for computing topological Hochschild homology of truncated Brown-Peterson spectra with certain coefficients. As a consequence, we show that $\mathbb{E}_3$-MU-algebra forms of truncated Brown-Peterson spectra $\mathrm{BP}\langle n\rangle$ are not Thom spectra at the prime $p=2$ for any $n\ge 2$.

math.AT↗
arXiv2026

Saturated directed spaces and generalized clocks

We construct a Hausdorff saturated directed space whose directed path space between two distinct points has nontrivial fundamental group although the corresponding trace space is a square. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from paths to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces whose compact subspaces are metrizable, these quotients are trivial q-fibrations. We construct a globular clock containing the directed circle and admitting regular maps from all q-cofibrant globular realizations, whereas a finite cellular example admits no regular map to the circle. Normalization also gives a proof of the quotient theorem for execution paths and establishes the homotopy lifting property for nondecreasing reparametrizations.

math.AT↗
arXiv2026

Non-simply connected Poincaré surgery

We give homotopy-theoretic proofs of the principal results of non-simply connected Poincaré surgery. These include the Fundamental Theorem of Poincaré surgery, the Poincaré transversality exact sequence, and the construction of Poincaré handle structures. All results apply in dimensions~$\ge 8$.

math.AT↗