[2003.06648] Flexible circuit of $d$ dimensional rigid matroid

Download the PDF of the paper entitled Flexible Circuits in $d$-Dimensional Rigid Matroids by Georg Grasegger and two other authors.

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overview: The bar joint framework $(G,p)$ of $\mathbb{R}^d$ is such that the only continuous movement that maintains the edge lengths of the vertices is the etc. If it arises from elongation, it is rigid. If $(G,p)$ is general, then its stiffness depends only on the underlying graph $G and is determined by the rank of the edge set of $G in general $d dimensions. is known. Rigid matroid $\mathcal{R}_d$. The complete combinatorial description of the rank function of this matroid is known for $d=1,2$ and all circuits in $\mathcal{R}_d$ are given by In the case of $\mathbb{R}^d$ it is generally meant to be a rigid body. Determining the rank function of d=1,2$. $\mathcal{R}_d$ is a long-standing open problem for $d\geq 3$ and A non-rigid circuit exists in $\mathcal{R}_d$. $ is the main factor why this problem is so hard. We begin our study of non-rigid circuits by characterizing $\mathcal{R}_d$ non-rigid circuits with at most $d+6$ vertices.