General position stresses

Authors: Robert Connelly, Steven J. Gortler, and Louis Theran
Preprint: 2305.12515, 2023
Full text: arXiv

Let \(G\) be a graph with \(n\) vertices, and \(d\) be a target dimension. In this paper we study the set of rank \(n−d−1\) matrices that are equilibrium stress matrices for at least one (unspecified) \(d\)-dimensional framework of \(G\) in general position. In particular, we show that this set is algebraically irreducible. Likewise, we show that the set of frameworks with such equilibrium stress matrices is irreducible. As an application, this leads to a new and direct proof that every generically globally rigid graph has a generic framework that is universally rigid.