Generic Rigidity of Graph Frameworks in Euclidean Space

TL;DR

This paper provides a combinatorial characterization of generic rigidity for bar-joint frameworks in all dimensions, extending the classical 2D results to higher dimensions using algebraic and geometric tools.

Contribution

It introduces a new combinatorial criterion for generic infinitesimal rigidity in all dimensions, unifying local and global conditions via Grassmannian relations.

Findings

Provides a combinatorial characterization valid in all dimensions.

Uses Grassmannian Plücker relations to verify rigidity.

Connects local Cramer's rule solutions with global rigidity conditions.

Abstract

The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer and independently in 1970 by Laman, analogous edge-density counts on subgraphs fail in higher dimensions. In this paper, we solve the problem by providing a combinatorial characterization of generic infinitesimal rigidity valid in all dimensions. By gluing together local versions of Cramer's rule at each vertex, we construct a globally valid self-stress on the edges. The compatibility conditions governing these local solutions are controlled by the Pl\"ucker relations on the Grassmannian $Gr(d+1, v)$, allowing us to check generic rigidity using the combinatorics of Young's straightening law on tableaux.