On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909522587549696 |
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| author | Lew, Alan Nevo, Eran Peled, Yuval Raz, Orit E. |
| author_facet | Lew, Alan Nevo, Eran Peled, Yuval Raz, Orit E. |
| contents | We define a generic rigidity matroid for $k$-volumes of a simplicial complex in $\mathbb{R}^d$, and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic $d$-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of $d$-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in $k$-faces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_01665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$ Lew, Alan Nevo, Eran Peled, Yuval Raz, Orit E. Combinatorics We define a generic rigidity matroid for $k$-volumes of a simplicial complex in $\mathbb{R}^d$, and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic $d$-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of $d$-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in $k$-faces. |
| title | On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.01665 |