Complex geometry of polygonal linkages

We present observations on the complex geometry of polygonal linkages arising in the framework of our approach to extremal problems on configuration spaces. Along with a few general remarks on applications of complex geometry and theory of residues, we present new results obtained in this way. Most of the new results are presented in the case of a planar quadrilateral linkage with generic lengths of the sides. First, we show that, for each configuration of planar quadrilateral linkage Q(a, b, c, d) with pairwise distinct side-lengths (a, b, c, d), the cross-ratio of its vertices belongs to the circle of radius ac/bd centered at the point $ 1\in \mathbb{C} $ . Next, we establish an analog of the Poncelet porism for a discrete dynamical system on a planar moduli space of a 4-bar linkage defined by the product of diagonal involutions and discuss some related issues suggested by a beautiful link to the theory of discrete integrable systems discovered by J. Duistermaat. We also present geometric results concerned with the electrostatic energy of point charges placed at the vertices of a quadrilateral linkage. In particular, we establish that all convex shapes of a quadrilateral linkage arise as the global minima of a system of charges placed at its vertices, and these shapes can be completely controlled by the value of the charge at just one vertex, which suggests a number of interesting problems. In conclusion, we describe a natural connection between certain extremal problems for configurations of linkage and convex polyhedra obtained from its configurations using the Minkowski 1897 theorem and present a few related remarks.