## Section: New Results

### On the maximal number of real embeddings of spatial minimally rigid graphs

Participants : Ioannis Emiris, Evangelos Bartzos.

In [15], we study the number of embeddings of
minimally rigid graphs in Euclidean space ${R}^{D}$, which is (by
definition) finite, modulo rigid transformations, for every generic
choice of edge lengths. Even though various approaches have been
proposed to compute it, the gap between upper and lower bounds is
still enormous. Specific values and its asymptotic behavior are
major and fascinating open problems in rigidity theory. Our work
considers the maximal number of real embeddings of minimally rigid
graphs in ${R}^{3}$. We modify a commonly used parametric semi-algebraic
formulation that exploits the Cayley-Menger determinant to minimize
the *a priori* number of complex embeddings, where the
parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in ${R}^{3}$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in ${R}^{3}$.

This is a joint work with J. Legersky (JK University, Linz, Austria) and E. Tsigaridas (PolSys, Inria).