Configuration spaces and signature formulas
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We show that nontrivial topological and geometric information about configuration spaces of linkages and tensegrities can be obtained using the signature formulas for the mapping degree and Euler characteristic. In particular, we prove that the Euler characteristics of such configuration spaces can be effectively calculated using signature formulas. We also investigate the critical points of signed area function on the configuration space of a planar polygon. We show that our approach enables one to effectively count the critical points in question and discuss a few related problems. One of them is concerned with the so-called cyclic polygons and formulas of Brahmagupta type. We describe an effective method of counting cyclic configurations of a given polygon and formulate four general conjectures about the critical points of the signed area function on the configuration space of a generic planar polygon. Several concrete results for planar quadrilaterals and pentagons are also presented.
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- 2.V. Arnold, A. Varchenko, and S. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Nauka, Moscow (2005).Google Scholar
- 3.G. Bibileishvili, “On configuration spaces of planar polygons and graphs,” Bull. Georgian Acad. Sci., 168, 203–206 (2003).Google Scholar
- 6.B. Dubrovin, S. Novikov, and A. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1988).Google Scholar
- 16.G. Khimshiashvili, “On configuration spaces of planar pentagons,” Zap. Nauch. Semin. POMI, 292, 120–129 (2002).Google Scholar
- 17.G. Khimshiashvili, “Circular configurations of planar polygons,” Bull. Georgian Natl. Acad. Sci., 176, No. 2, 27–30 (2008).Google Scholar
- 19.A. Pugh, An Introduction to Tensegrity, Univ. Calif. Press, Berkeley (1976).Google Scholar