The number of realisations of a rigid graph in Euclidean and spherical geometries
By extending the concept of graph rigidity to allow for complex solutions, the number of edge-length equivalent realisations of a d-dimensional framework can be seen to be a generic property (in that it is constant over a non-empty Zariski open set of realisations) for both Euclidean and spherical geometries: these two numbers are now said to be the d-realisation number and the spherical d-realisation number of the graph respectively. Recently developed algorithms for computing realisation numbers have observed that for minimally 2-rigid graphs with 12 vertices or less, the 2-realisation number is always less than or equal to the spherical 2-realisation number; for example, the 3-prism has a 2-realisation number of 12 and a spherical 2-realisation number of 16. In this talk we confirm that this observation is indeed true for any graph in any dimension: specifically, for any dimension d, the d-realisation number of a graph is always at most the spherical d-realisation number of the same graph. This result is obtained via new techniques involving coning. This result is joint work with Georg Grasegger.