Flexible circuits in the $d$-dimensional rigidity matroid

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 $dgeq 3$, and the existence of non-rigid circuits in $mathcal{R}_d$ for $dgeq 3$ 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.