Flexible circuits in the $d$-dimensional rigidity matroid

Abstract

A bar-joint framework $(G,p)$ in ${\mathbb{R}}^d$ is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of ${\mathbb{R}}^d$. It is known that, when $(G,p)$ is generic, its rigidity depends only on the underlying graph $G$, and is determined by the rank of the edge set of $G$ in the generic $d$-dimensional rigidity matroid ${\mathcal{R}}_d$. Complete combinatorial descriptions of the rank function of this matroid are known when $d=1,2$, and imply that all circuits in ${\mathcal{R}}_d$ are generically rigid in ${\mathbb{R}}^d$ when $d=1,2$. Determining the rank function of ${\mathcal{R}}_d$ is a long standing open problem when $d\ge 3$, and the existence of non-rigid circuits in ${\mathcal{R}}_d$ for $d\ge 3$ is a major contributing factor to why this problem is so difficult. We begin a study of non-rigid circuits by characterising the non-rigid circuits in ${\mathcal{R}}_d$ which have at most $d+6$ vertices.

21 pages, 6 figures. Final version, with a short corrigendum appended to the end which gives counterexamples to Lemma 18(a) and Conjecture 17