Geodesic
Topology of random linkages
Farber, Michael1
1Department of Mathematical Sciences, University of Durham, UK
Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 155-171
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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 ℝ3. We also prove results about higher moments of Betti numbers.

DOI : 10.2140/agt.2008.8.155
Keywords: linkage, polygon space, random manifold, betti number

Farber, Michael  1

1 Department of Mathematical Sciences, University of Durham, UK 
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Farber, Michael. Topology of random linkages. Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 155-171. doi: 10.2140/agt.2008.8.155

[1] M M Ali, Content of the frustum of a simplex, Pacific J. Math. 48 (1973) 313

[2] H B Curry, I J Schoenberg, On Pólya frequency functions. IV. The fundamental spline functions and their limits, J. Analyse Math. 17 (1966) 71

[3] M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211

[4] M Farber, J C Hausmann, D Schütz, On the conjecture of Kevin Walker, preprint (2007)

[5] M Farber, T Kappeler, Betti numbers of random manifolds, to appear in Homology, Homotopy and Applications

[6] M Farber, D Schütz, Homology of planar polygon spaces, Geom. Dedicata 125 (2007) 75

[7] J C Hausmann, A Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998) 281

[8] J C Hausmann, E Rodriguez, The space of clouds in Euclidean space, Experiment. Math. 13 (2004) 31

[9] M Kapovich, J Millson, On the moduli space of polygons in the Euclidean plane, J. Differential Geom. 42 (1995) 133

[10] D G Kendall, D Barden, T K Carne, H Le, Shape and shape theory, Wiley Series in Probability and Statistics, John Wiley Sons Ltd. (1999)

[11] A A Klyachko, Spatial polygons and stable configurations of points in the projective line, from: "Algebraic geometry and its applications (Yaroslavl', 1992)", Aspects Math., E25, Vieweg (1994) 67

[12] G Varsi, The multidimensional content of the frustum of the simplex, Pacific J. Math. 46 (1973) 303

[13] K Walker, Configuration spaces of linkages, Undergraduate thesis, Princeton University (1985)

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