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

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 $L_1$.