超立方图的Vietoris-Rips复形(尺度3

Samir Shukla
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引用次数: 8

摘要

对于度量空间$(X, d)$和尺度参数$r \geq 0$, Vietoris-Rips复形$\mathcal{VR}(X;r)$是顶点集$X$上的简单复形,其中有限集$\sigma \subseteq X$是单纯形当且仅当$\sigma$的直径不大于$r$。对于$n \geq 1$,设$\mathbb{I}_n$表示$n$维超立方图。本文证明了$\mathcal{VR}(\mathbb{I}_n;r)$仅在$4$和$7$维上具有非平凡的约简同调。因此,我们回答了Adamaszek和Adams最近提出的一个问题。一个(有限)简单复合体$\Delta$是$d$ -可折叠的,如果它可以通过重复移除包含在$\Delta$的唯一最大面中大小最多为$d$的面而简化为空洞复合体。$\Delta$的可折叠数是使$\Delta$为$d$ -可折叠的最小整数$d$。我们证明$\mathcal{VR}(\mathbb{I}_n;r)$对于$r \in \{2, 3\}$的可折叠性数为$2^r$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Vietoris-Rips Complexes (with Scale 3) of Hypercube Graphs
For a metric space $(X, d)$ and a scale parameter $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $\sigma \subseteq X$ is a simplex if and only if diameter of $\sigma$ is at most $r$. For $n \geq 1$, let $\mathbb{I}_n$ denotes the $n$-dimensional hypercube graph. In this paper, we show that $\mathcal{VR}(\mathbb{I}_n;r)$ has non trivial reduced homology only in dimensions $4$ and $7$. Therefore, we answer a question posed by Adamaszek and Adams recently. A (finite) simplicial complex $\Delta$ is $d$-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most $d$ that is contained in a unique maximal face of $\Delta$. The collapsibility number of $\Delta$ is the minimum integer $d$ such that $\Delta$ is $d$-collapsible. We show that the collapsibility number of $\mathcal{VR}(\mathbb{I}_n;r)$ is $2^r$ for $r \in \{2, 3\}$.
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