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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.

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References

  1. A. Agrachev and R. Gamkrelidze, “Quadratic mapping and smooth vector-functions: Euler characteristic of the level set,” Itogi Nauki Techn. Sovr. Probl. Mat. Noveishie Dostizhenia, 35, 179–239 (1989).

    MATH  MathSciNet  Google Scholar 

  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 

  4. J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin–Heidelberg–New York (1998).

    MATH  Google Scholar 

  5. J. Bruce, “Euler characteristics of real varieties,” Bull. London Math. Soc., 22, 213–219 (1990).

    MathSciNet  Google Scholar 

  6. B. Dubrovin, S. Novikov, and A. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  7. D. Eisenbud and H. Levine, “An algebraic formula for the degree of a C ∞ map germ,” Ann. Math., 106, 19–44 (1977).

    Article  MathSciNet  Google Scholar 

  8. M. Fedorchuk and I. Pak, “Rigidity and polynomial invariants of convex polytopes,” Duke Math. J., 129, No. 2, 371–404 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Giorgadze, “Quadratic mappings and configuration spaces,” Banach Center Publ., 62, 159–172 (2004).

    MathSciNet  Google Scholar 

  10. M. Hirsch, Differential Topology, Springer-Verlag, Berlin–Heidelberg–New York (1976).

    MATH  Google Scholar 

  11. Y. Kamiyama, “The Euler characteristic of the moduli space of polygons in higher-dimensional Euclidean space,” Kyushu J. Math., 54, 333–369 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Kapovich and J. Millson, “Universality theorems for configuration spaces of planar linkages,” Topology, 41, 1051–1107 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Khimshiashvili, “On the local degree of a smooth mapping,” Bull. Acad. Sci. Georgian SSR, 85, 309–312 (1977).

    MATH  Google Scholar 

  14. G. Khimshiashvili, “On the cardinality of a semi-algebraic subset,” Georgian Math. J., 1, 111–120 (1994).

    Article  Google Scholar 

  15. G. Khimshiashvili, “Signature formulae for topological invariants,” Proc.A. Razmadze Math. Inst., 125, 1–121 (2001).

    MATH  MathSciNet  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 

  18. A. Lecki and Z. Szafraniec, “An algebraic method for calculating the topological degree,” Banach Center Publ., 35, 73–83 (1996).

    MathSciNet  Google Scholar 

  19. A. Pugh, An Introduction to Tensegrity, Univ. Calif. Press, Berkeley (1976).

    Google Scholar 

  20. D. Robbins, “Areas of polygons inscribed in a circle,” Discr. Comput. Geom., 12, 223–236 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  21. W. Thurston, “Shapes of polyhedra and triangulations of the sphere,” Geom. Topol. Monogr., 1, 511–549 (1998).

    Article  MathSciNet  Google Scholar 

  22. V. Varfolomeev, “Inscribed polygons and Heron polynomials,” Sb. Math., 194, 311–321 (2003).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. I. Tchavtchavadze State University, A. Razmadze Mathematical Institute, Tbilisi, Georgia

    G. Khimshiashvili

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

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Khimshiashvili, G. Configuration spaces and signature formulas. J Math Sci 160, 727–736 (2009). https://doi.org/10.1007/s10958-009-9524-x

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  • DOI: https://doi.org/10.1007/s10958-009-9524-x

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