Persistent Combinatorial Model of the Restricted Second Configuration Space of Metric Star Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911475766919168 |
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| author | Li, Wenwen Ozaydin, Murad |
| author_facet | Li, Wenwen Ozaydin, Murad |
| contents | In this work, we present explicit constructions and computations of representative cycles for a nontrivial 2-parameter persistence module arising from the configuration space of metric star graphs. For all edge-length vector $\mathbf{L}=(L_1, L_2, \dots, L_k)\in(\mathbb{R}_{>0})^k$, we construct a bipartite weighted graph $(G_k)_{\mathbf{L}}$ and define filtering functions on the set of vertices and set of edges of $(G_k)_{\mathbf{L}}$ to obtain a filtration (denoted by $(G_k)_{-,\mathbf{L}}$) consisting of geometric realization of subgraphs of $(G_k)_{\mathbf{L}}$. We show that such a filtration is naturally isomorphic to the filtration of the restricted second configuration space of metric star graphs $(\mathsf{Star}_k)^2_{r,\mathbf{L}}$ concerning the restraint parameter $r$ and an (arbitrary but fixed) edge-length vector $\mathbf{L}$. Additionally, we show that the filtration $(G_k)_{-,\mathbf{L}}$ is compatible with the edge-length vector $\mathbf{L}$ up to isotopy, establishing an equivalence between the associated $(k+1)$-parameter persistence modules $PH_{i}((\mathsf{Star}_k)^2_{-,-};\mathbb{F})$ and $PH_{i}((G_k)_{-,-};\mathbb{F})$. We call the (multi-)filtration $(G_k)_{-,-}$ a \textit{persistent combinatorial model} of the multifiltration $(\mathsf{Star}_k)^2_{-,-}$. Using this model, we construct explicit compatible cycle representatives for $PH_{1}((\mathsf{Star}_k)^2_{-,-};\mathbb{F})$ in the bifiltration obtained by fixing $L_2, \dots, L_k > 0$ and varying only $r$ and $L_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Persistent Combinatorial Model of the Restricted Second Configuration Space of Metric Star Graphs Li, Wenwen Ozaydin, Murad Algebraic Topology 55N31 In this work, we present explicit constructions and computations of representative cycles for a nontrivial 2-parameter persistence module arising from the configuration space of metric star graphs. For all edge-length vector $\mathbf{L}=(L_1, L_2, \dots, L_k)\in(\mathbb{R}_{>0})^k$, we construct a bipartite weighted graph $(G_k)_{\mathbf{L}}$ and define filtering functions on the set of vertices and set of edges of $(G_k)_{\mathbf{L}}$ to obtain a filtration (denoted by $(G_k)_{-,\mathbf{L}}$) consisting of geometric realization of subgraphs of $(G_k)_{\mathbf{L}}$. We show that such a filtration is naturally isomorphic to the filtration of the restricted second configuration space of metric star graphs $(\mathsf{Star}_k)^2_{r,\mathbf{L}}$ concerning the restraint parameter $r$ and an (arbitrary but fixed) edge-length vector $\mathbf{L}$. Additionally, we show that the filtration $(G_k)_{-,\mathbf{L}}$ is compatible with the edge-length vector $\mathbf{L}$ up to isotopy, establishing an equivalence between the associated $(k+1)$-parameter persistence modules $PH_{i}((\mathsf{Star}_k)^2_{-,-};\mathbb{F})$ and $PH_{i}((G_k)_{-,-};\mathbb{F})$. We call the (multi-)filtration $(G_k)_{-,-}$ a \textit{persistent combinatorial model} of the multifiltration $(\mathsf{Star}_k)^2_{-,-}$. Using this model, we construct explicit compatible cycle representatives for $PH_{1}((\mathsf{Star}_k)^2_{-,-};\mathbb{F})$ in the bifiltration obtained by fixing $L_2, \dots, L_k > 0$ and varying only $r$ and $L_1$. |
| title | Persistent Combinatorial Model of the Restricted Second Configuration Space of Metric Star Graphs |
| topic | Algebraic Topology 55N31 |
| url | https://arxiv.org/abs/2603.00914 |