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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Morse index of a cyclic polygon. II


Author: A. Zhukova
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 461-474
MSC (2010): Primary 14M06; Secondary 53D30, 58E05
Published electronically: March 21, 2013
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Abstract: A polygonal linkage can be imagined as a set of $ n$ rigid bars connected by links cyclically. This construction lies on a plane and can rotate freely around the links, with allowed self-intersections. On the moduli space of the polygonal linkage, the signed area function $ A$ is defined. G. Panina and G. Khimshiashvili proved that cyclic configurations of a polygonal linkage are the critical points of $ A$. Later, G. Panina and the author described a way to compute the Morse index of a cyclic configuration of a polygonal linkage. In this paper a simple formula for the Morse index of a cyclic configuration is given. Also, a description is obtained for all possible local extrema of $ A$.


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Additional Information

A. Zhukova
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: millionnaya13@ya.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01247-7
PII: S 1061-0022(2013)01247-7
Keywords: Linkages, moduli space, Morse theory
Received by editor(s): May 29, 2011
Published electronically: March 21, 2013
Additional Notes: Partially supported by the program “Research in Pairs” of Mathematisches Forschungsinstitut Oberwolfach in 2010. The author thanks G. Panina, G. Khimshiashvili, and D. Siersma for their help and useful remarks
Article copyright: © Copyright 2013 American Mathematical Society