On the geometry and topology of moduli spaces of multi-polygonal linkages

The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal linkage by taking two free linkages and identifying initial and terminal vertices. This can be generalized so that one takes three free linkages and identifies initial and terminal vertices. Then one obtains a linkage which contains multiple polygons, any two of which have shared edges. The geometric and topological properties of moduli spaces of these multi-polygonal linkages are studied. These spaces turn out to be compact algebraic varieties. Multi-quadrilateral linkages whose moduli spaces are at most one dimensional are classified. The dimensions and some Euler characteristics are computed, and conditions under which these spaces are smooth manifolds are determined. Some conditions are also given for when the moduli spaces are connected and when they are disjoint unions of two moduli spaces of polygonal linkages

Generic singular configurations of linkages

We study the topological and differentiable singularities of the configuration space C(\Gamma) of a mechanical linkage \Gamma in d-dimensional Euclidean space, defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical interpretation, and give an example of a type of mechanism for which this criterion identifies all singularities

Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the "Simplicial Steinitz problem". It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets"

Cross Ratios of Quadrilateral Linkages

We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we describe the image of cross-ratio map for quadrilateral linkage and planar robot 3-arm.Comment: 12 pages,9 figures. March 2014: Proofs are added, typo's corrected; March 2015: Typo's correcte