Flexible circuits in the $d$-dimensional rigidity matroid
| Autor: | Grasegger, Georg, Guler, Hakan, Jackson, Bill, Nixon, Anthony |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1002/jgt.22780, |
| Popis: | 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. Comment: 21 pages, 6 figures. Final version, with a short corrigendum appended to the end which gives counterexamples to Lemma 18(a) and Conjecture 17 |
| Databáze: | arXiv |
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