Orientation-Reversing Crystallographic Rigidity

This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group ${\mathbb{Z}}^2⋊{\mathcal{C}}_s$, also known as $pm$ in crystallography, under a fixed lattice representation. Corresponding results for the wallpaper groups $cm$ and $pg$ follow directly from this. The method used also provides an inductive construction for the corresponding gain graphs, in terms of Henneberg-type graph operations.