Isostatic Block and Hole Frameworks

Abstract: A longstanding problem in rigidity theory is to characterize the graphs which are minimally generically rigid in 3-space. The results of Cauchy, Dehn, and Alexandrov give one important class: the triangulated convex spheres, but there is an ongoing desire for further classes. We provide such a class, along with methods to verify generic rigidity that can be extended to other classes. These methods are based on a controlled sequence of vertex splits, a graph theoretic operation known to take a minimally generically rigid framework to a new minimally generically rigid framework with one more vertex. One motivation for this is to have well-understood frameworks which can be used to explore Mathematical Allostery - frameworks in which adding bars at one site, causes changes in rigidity at a distant site. This is an initial step in exploring the possibility of mechanical models for an important behaviour in proteins.