On the area of constrained polygonal linkages

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova)

Simple game induced manifolds

Starting by a simple game $Q$ as a combinatorial data, we build up a cell complex $M(Q)$, whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{ simple game induced manifold.} By some motivations coming from polygonal linkages, we think of $Q$ and of $M(Q)$ as of\textit{ a quasilinkage} and the \textit{moduli space of the quasilinkage} respectively. We present some examples of quasilinkages and show that the moduli space retains many properties of moduli space of polygonal linkages. In particular, we show that the moduli space $M(Q)$ is homeomorphic to the space of stable point configurations on $S^1$, for an associated with a quasilinkage notion of stability