论文标题
$ d $维刚性刚性的灵活电路
Flexible circuits in the $d$-dimensional rigidity matroid
论文作者
论文摘要
$ \ mathbb {r}^d $中的bar-cont框架$（g，p）$是刚性的，如果唯一的边缘长度保留顶点的连续运动来自$ \ mathbb {r}^d $的同密度。众所周知，当$（g，p）$是通用的时候，其刚度仅取决于基础图$ g $，并且由$ g $的边缘集合在通用$ d $ d $ d $二维的刚性刚性$ \ mathcal $ \ mathcal {r} _d $中确定。当$ d = 1,2 $时，已知对该矩阵的等级函数的完整组合描述，这意味着$ \ mathcal {r} _d $中的所有电路通常在$ \ mathbb {r}^d $中固定在$ d = 1,2 $时。当$ d \ geq 3 $以及在$ \ Mathcal {r} _d $中存在非韧性电路时，确定$ \ Mathcal {r} _d $的排名函数是一个长期的开放问题，对于$ \ natcal {r} _d $ for $ d \ geq 3 $是如此困难的原因。我们通过表征$ \ Mathcal {r} _d $中的非刚性电路来开始对非刚性电路的研究，最多具有$ d+6 $ dertices。
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 $d\geq 3$, and the existence of non-rigid circuits in $\mathcal{R}_d$ for $d\geq 3$ 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.
