Missing faces of neighborly and nearly neighborly polytopes and spheres
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915507524861952 |
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| author | Novik, Isabella Zheng, Hailun |
| author_facet | Novik, Isabella Zheng, Hailun |
| contents | For a $(d-1)$-dimensional simplicial complex $Δ$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $Δ$ and $m_i$ be the number of missing $i$-faces of $Δ$. In the nineties, Kalai asked for a characterization of the $m$-numbers of simplicial polytopes and spheres -- a problem that remains wide open to this day. Here, we study the $m$-numbers of nearly neighborly and neighborly polytopes and spheres. Specifically, for $d\geq 4$, we obtain a lower bound on $m_{\lfloor d/2\rfloor}$ in terms of $f_0$ and $f_{\lfloor d/2\rfloor-1}$ in the class of all $(\lfloor d/2\rfloor-1)$-neighborly $(d-1)$-spheres. For neighborly spheres, we (almost) characterize the $m$-numbers of $2$-neighborly $4$-spheres, and we show that, for all odd values of $k$, there exists an infinite family of neighborly simplicial $2k$-spheres with $m_{k+1}=0$. Along the way, we provide a simple numerical condition based on the $m$-numbers that allows to establish non-polytopality of some neighborly odd-dimensional spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Missing faces of neighborly and nearly neighborly polytopes and spheres Novik, Isabella Zheng, Hailun Combinatorics 52B05, 05E45 For a $(d-1)$-dimensional simplicial complex $Δ$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $Δ$ and $m_i$ be the number of missing $i$-faces of $Δ$. In the nineties, Kalai asked for a characterization of the $m$-numbers of simplicial polytopes and spheres -- a problem that remains wide open to this day. Here, we study the $m$-numbers of nearly neighborly and neighborly polytopes and spheres. Specifically, for $d\geq 4$, we obtain a lower bound on $m_{\lfloor d/2\rfloor}$ in terms of $f_0$ and $f_{\lfloor d/2\rfloor-1}$ in the class of all $(\lfloor d/2\rfloor-1)$-neighborly $(d-1)$-spheres. For neighborly spheres, we (almost) characterize the $m$-numbers of $2$-neighborly $4$-spheres, and we show that, for all odd values of $k$, there exists an infinite family of neighborly simplicial $2k$-spheres with $m_{k+1}=0$. Along the way, we provide a simple numerical condition based on the $m$-numbers that allows to establish non-polytopality of some neighborly odd-dimensional spheres. |
| title | Missing faces of neighborly and nearly neighborly polytopes and spheres |
| topic | Combinatorics 52B05, 05E45 |
| url | https://arxiv.org/abs/2505.20699 |