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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2604.05442 |
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| _version_ | 1866910145435402240 |
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| author | Heaton, Alexander |
| author_facet | Heaton, Alexander |
| contents | The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer and independently in 1970 by Laman, analogous edge-density counts on subgraphs fail in higher dimensions. In this paper, we solve the problem by providing a combinatorial characterization of generic infinitesimal rigidity valid in all dimensions. By gluing together local versions of Cramer's rule at each vertex, we construct a globally valid self-stress on the edges. The compatibility conditions governing these local solutions are controlled by the Plücker relations on the Grassmannian $Gr(d+1, v)$, allowing us to check generic rigidity using the combinatorics of Young's straightening law on tableaux. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05442 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generic Rigidity of Graph Frameworks in Euclidean Space Heaton, Alexander Combinatorics 52C25 The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer and independently in 1970 by Laman, analogous edge-density counts on subgraphs fail in higher dimensions. In this paper, we solve the problem by providing a combinatorial characterization of generic infinitesimal rigidity valid in all dimensions. By gluing together local versions of Cramer's rule at each vertex, we construct a globally valid self-stress on the edges. The compatibility conditions governing these local solutions are controlled by the Plücker relations on the Grassmannian $Gr(d+1, v)$, allowing us to check generic rigidity using the combinatorics of Young's straightening law on tableaux. |
| title | Generic Rigidity of Graph Frameworks in Euclidean Space |
| topic | Combinatorics 52C25 |
| url | https://arxiv.org/abs/2604.05442 |