2-Parameter Persistent Homology of the Second Configuration Spaces of Star Graphs

Abstract: In this talk, I will provide a brief introduction to the restricted second configuration space of a metric graph $X_{\mathbf{L}}$, denoted by $X_{\mathbf{L}}^2$, which depends on the restraint parameter $r$ and edge length vector $\mathbf{L}$. I will then present joint work with Murad Özaydın on $2$-parameter persistence modules $PH_i(X_{\mathbf{L}};\mathbb{F})$ when $X_{\mathbf{L}}$ is a metric star graph and the edge length vector $\mathbf{L}=(L,1,\dots,1)$ for $i=0$ or $1$ . We demonstrate that $PH_i(X_{\mathbf{L}};\mathbb{F})$ is isomorphic to a $2$-parameter persistence module $N$ where its restriction to each chamber of the parameter space (of $X_{\mathbf{L}}$) is a constant persistence module. Moreover, we provide the indecomposable direct summands for these persistence modules.

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