Homotopy equivalence of digital pictures in $\mathbb{Z}^2$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Lee, Dae-Woong Staecker, P. Christopher |
| author_facet | Lee, Dae-Woong Staecker, P. Christopher |
| contents | We investigate the properties of digital homotopy in the context of digital pictures $(X,κ,\bar κ)$, where $X\subsetneq \Z^n$ is a finite set, $κ$ is an adjacency relation on $X$, and $\bar κ$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\Z^2$. We define a numerical homotopy-type invariant for digital pictures in $\Z^2$ called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by ``filling in the gaps'' of any row or column. We show that a digital picture $(X,c_i,c_j)$ is homotopy equivalent to only finitely many other digital pictures $(Y,c_i,c_j)$. At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy equivalence of digital pictures in $\mathbb{Z}^2$ Lee, Dae-Woong Staecker, P. Christopher Algebraic Topology Discrete Mathematics Primary 68U03, Secondary 55P15, 55P99, 54H30 We investigate the properties of digital homotopy in the context of digital pictures $(X,κ,\bar κ)$, where $X\subsetneq \Z^n$ is a finite set, $κ$ is an adjacency relation on $X$, and $\bar κ$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\Z^2$. We define a numerical homotopy-type invariant for digital pictures in $\Z^2$ called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by ``filling in the gaps'' of any row or column. We show that a digital picture $(X,c_i,c_j)$ is homotopy equivalent to only finitely many other digital pictures $(Y,c_i,c_j)$. At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture. |
| title | Homotopy equivalence of digital pictures in $\mathbb{Z}^2$ |
| topic | Algebraic Topology Discrete Mathematics Primary 68U03, Secondary 55P15, 55P99, 54H30 |
| url | https://arxiv.org/abs/2509.03023 |