Abstract
Algebraic formulas for the Euler characteristic are applied to the topological study of the moduli spaces of planar polygons. In particular, a complete description of the moduli spaces of planar pentagons is obtained by using a computer algorithm for the calculation of the Euler characteristic of algebraic varieties. Bibliography: 17 titles.
REFERENCES
V. Alexandrov, “Implicit function theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks”, Monatshefte Math., 132, 269–288 (2001).
V. Arnold, A. Varchenko, and S. Gusein-zade, Singularities of Differentiable Mappings [in Russian], Vol. 1, Moscow (1982).
W. Bruce, “Euler characteristics of real varieties”, Bull. Lond. Math. Soc., 22, 547–552 (1990).
T. Delzant, “Hamiltoniens periodiques et images convexes de l’application moment”, Bull. Soc. Math. France, 116, 315–339 (1988).
D. Eisenbud and H. Levine, “An algebraic formula for the degree of a C∞ map germ”, Ann. Math., 106, 19–44 (1977).
C. Gibson and P. Newstead, “On the geometry of the planar 4-bar mechanism”, Acta Appl. Math., 7, 113–135 (1986).
F. X. Grimm, “Konfigurationsräaume von Graphen”, Diplomarbeit, Regensburg Univ. (1991).
M. Kapovich and J. Millson, “On the moduli spaces of polygons in the Euclidean plane”, J. Diff. Geom., 42, 133–164 (1995).
M. Kapovich and J. Millson, “The symplectic geometry of polygons in Euclidean space”, J. Diff. Geom., 44, 479–513 (1996).
G. Khimshiashvili, “On the local degree of a smooth mapping”, Bull. Akad. Nauk Gruz. SSR, 85, 309–312 (1977).
G. Khimshiashvili, “On the cardinality of a semi-algebraic subset”, Georgian Math. J., 1, 277–286 (1994).
G. Khimshiashvili, “Signature formulae for topological invariants”, Tbilisi Math. Inst. Proc., 125, 1–121 (2001).
A. Khovansky, Fewnomials, AMS Publ., Providence (1991).
A. Lecki and Z. Szafraniec, “An algebraic method for calculating the topological degree”, Banach Center Publ., 35, 73–83 (1996).
M. Shakhinpoor, Robot Engineering Textbook, New Mexico Univ. (1987).
A. Szücs (oral communication).
K. Walker, Configuration Spaces of Linkages, Undergraduate thesis, Princeton Univ. (1985).
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 120–129.
Rights and permissions
About this article
Cite this article
Khimshiashvili, G. On configuration spaces of planar pentagons. J Math Sci 126, 1111–1116 (2005). https://doi.org/10.1007/s10958-005-0100-8
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10958-005-0100-8