, Volume 160, Issue 6, pp 727-736
Date: 22 Jul 2009

Configuration spaces and signature formulas

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Abstract

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.

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.