Rigidity theory for matroids

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in ℝ^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this wider setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial), and have little to do with the special nature of graphic matroids or the field R. Our main result is a "nesting theorem" relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence. The number of points on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.