Topology of random linkages

Farber, Michael

doi:10.2140/agt.2008.8.155

Cited by 17 publications

(17 citation statements)

References 14 publications

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“…The complexity of motion planning algorithms is measured by a numerical invariant TC(X) which depends on the homotopy type of the configuration space X of the system [17]. This invariant is defined as the Schwarz genus (also known as the "sectional category") of the path-space fibration…”

Section: Motion Planning Algorithms and The Concept Of Topological Comentioning

confidence: 99%

“…The invariant TC(X) admits an upper bound in terms of the dimension of the configuration space X, TC(X) ≤ 2 dim(X) + 1 (10) see [17], Theorem 4. There are many examples when inequality (10) is sharp: take for instance X = T n ♯T n , the connected sum of two copies of a torus, having the topological complexity TC(X) = 2n + 1.…”

Section: Motion Planning Algorithms and The Concept Of Topological Comentioning

confidence: 99%

“…Configuration spaces of mechanical linkages with random bar lengths were studied in papers [17], [18]. These are closed smooth manifolds depending on a large number of independent random parameters.…”

Section: Introductionmentioning

confidence: 99%

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Topology of Random Right Angled Artin Groups

Costa

Farber

2011

J. Topol. Anal.

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Abstract. In this paper we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values, when n → ∞. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.

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Section: Motion Planning Algorithms and The Concept Of Topological Comentioning

confidence: 99%

Section: Motion Planning Algorithms and The Concept Of Topological Comentioning

confidence: 99%

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Topology of Random Right Angled Artin Groups

Costa

Farber

2011

J. Topol. Anal.

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“…In a subsequent work we shall describe a generalization of Theorem 1.1 which allows the dimension p to grow with n. See also [3] where the average Betti numbers of polygon spaces in R 3 are calculated. Paper [3] also contains results for more general probability measures which explain the "universality phenomenon".…”

Section: Introductionmentioning

confidence: 99%

“…Paper [3] also contains results for more general probability measures which explain the "universality phenomenon".…”

Section: Introductionmentioning

confidence: 99%

Betti numbers of random manifolds

Farber

Kappeler

2008

Homology, Homotopy and Applications

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We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof we use a combination of geometric and analytic tools. The average Betti numbers are expressed in terms of volumes of intersections of a simplex with certain half-spaces.

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Some remarks on Betti numbers of random polygon spaces

Dombry

Mazza

2009

Random Struct. Alg.

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Polygon spaces such as, or the three-dimensional analogs N play an important rôle in geometry and topology, and are also of interest in robotics where the l i model the lengths of robot arms. When n is large, one can assume that each l i is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincaré polynomial. We study the average values of Betti numbers of dimension p n when p n → ∞ as n → ∞. We also focus on the limiting mean Poincaré polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where l i ≡l. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincaré polynomials.

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Topology of random linkages

Cited by 17 publications

References 14 publications

Topology of Random Right Angled Artin Groups

Topology of Random Right Angled Artin Groups

Betti numbers of random manifolds

Some remarks on Betti numbers of random polygon spaces

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