Morse index of a cyclic polygon. II

Abstract

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.

Oriented area as a Morse function on polygon spaces

“…The formula more or less explicitly appeared in [6,9], and [11], but in this form, with the precise condition of being Morse, the theorem was proved only in [3]. The following definition was first given in [3].…”

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

“…Generically, L is a smooth closed manifold [1,2], and the oriented area A is a Morse function on L. Before we recall a formula for the Morse index of a cyclic configuration from [6], [4], let us fix the following notation for a cyclic polygon P . ω P = w(P, O) is the winding number of P with respect to the center O of the circumscribed circle.…”

“…• L is a smooth closed manifold whose diffeomorphic type depends on the edge lengths [1,2]. • The oriented area A is a Morse function whose critical points are cyclic configurations (that is, polygons with all the vertices lying on a circle), whose Morse indices are known, see Theorem 2, [3,5,6]. The Morse index depends not only on the combinatorics of a cyclic polygon, but also on some metric data.…”

On the area of constrained polygonal linkages

“…We recall a short formula for Morse index of a cyclic configuration from [12], [7]. We fix the following notation for a cyclic configuration.…”

“…One thinks of it as of a flexible polygon with rigid edges and revolving joints at the vertices whose ambient space is the Euclidean plane. The idea of considering the oriented area as a Morse function on its configuration space has already led to some non-trivial results: the critical points (or, equivalently, critical configurations) are easily describable, and there exists a short formula for the Morse index [5], [7], [8], [12]. In some further generalization [9] the oriented area proves to be an exact Morse function.…”