Persistent Combinatorial Model of the Restricted Second Configuration Space of Metric Star Graphs

TL;DR

This paper develops a combinatorial graph-based model to analyze the persistent homology of the configuration space of metric star graphs, providing explicit cycle representatives and establishing isomorphisms between different filtrations.

Contribution

It introduces a persistent combinatorial model that simplifies the study of the second configuration space of metric star graphs and constructs explicit cycle representatives.

Findings

Constructed a bipartite weighted graph model $(G_k)_{L}$ for the configuration space.

Proved the model's filtration is isomorphic to the configuration space filtration.

Explicitly constructed compatible cycle representatives for the first homology.

Abstract

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 $L_1$.