It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration.It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.

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.

We work with discrete Morse theory. Let F be a discrete Morse function on a simplicial complex L. We construct a discrete Morse function ∆(F ) on the barycentric subdivision ∆(L). The constructed function ∆(F ) "behaves the same way" as F , i. e. has the same number of critical simplexes and the same gradient path structure.

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set [n] = {1, . . . , n}. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set [n + 1]. The cyclopermutohedron was introduced by the second author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (a) the volume of the cyclopermutohedron equals zero, and (b) the homology groups H k for k = 0, . . . , n −2 of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration. MSC:52Cxx, 58E05