Computing 1-Periodic Persistent Homology with Finite Windows
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916955176304640 |
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| author | Onus, Adam Skraba, Primoz |
| author_facet | Onus, Adam Skraba, Primoz |
| contents | Let $K$ be a periodic cell complex endowed with a covering $q:K\to G$ where $G$ is a finite quotient space of equivalence classes under translations acting on $K$. We assume $G$ is embedded in a space whose homotopy type is a $d$-torus for some $d$, which introduces "toroidal cycles" in $G$ which do not lift to cycles in $K$ by $q$ . We study the behaviour of toroidal and non-toroidal cycles for the case $K$ is 1-periodic, i.e. $G=K/\mathbb{Z}$ for some free action of $\mathbb{Z}$ on $K$. We show that toroidal cycles can be entirely classified by endomorphisms on the homology of unit cells of $K$, and moreover that toroidal cycles have a sense of unimodality when studying the persistent homology of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00709 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Computing 1-Periodic Persistent Homology with Finite Windows Onus, Adam Skraba, Primoz Algebraic Topology 55N31, 57Z25, 55-08 Let $K$ be a periodic cell complex endowed with a covering $q:K\to G$ where $G$ is a finite quotient space of equivalence classes under translations acting on $K$. We assume $G$ is embedded in a space whose homotopy type is a $d$-torus for some $d$, which introduces "toroidal cycles" in $G$ which do not lift to cycles in $K$ by $q$ . We study the behaviour of toroidal and non-toroidal cycles for the case $K$ is 1-periodic, i.e. $G=K/\mathbb{Z}$ for some free action of $\mathbb{Z}$ on $K$. We show that toroidal cycles can be entirely classified by endomorphisms on the homology of unit cells of $K$, and moreover that toroidal cycles have a sense of unimodality when studying the persistent homology of $G$. |
| title | Computing 1-Periodic Persistent Homology with Finite Windows |
| topic | Algebraic Topology 55N31, 57Z25, 55-08 |
| url | https://arxiv.org/abs/2312.00709 |