On the existence of paradoxical motions of generically rigid graphs on the sphere
This addresses a theoretical problem in geometric rigidity for mathematicians, offering incremental insights into spherical flexibility.
The paper tackles the problem of characterizing which graphs can flex on the sphere without collapsing, providing a combinatorial condition based on edge colorings and necessary length relations, and classifies all motions for a specific 3+3 bipartite graph.
We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with $3+3$ vertices where no two vertices coincide or are antipodal.