Persistent Homology of Configuration Spaces of Trees

Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the minimum distance allowed between each pair of robots) can be interpreted as a collection of all possible configurations of two robots moving in $(X,\delta)$. In this project, we study the $2$-parameter persistence modules associated with the second configuration spaces of the star graph ($\mathsf{Star}_k$, $k\geq 3$) and the generalized H-graph ($\mathcal{H}_{m,n}$, $m,n\geq 3$) with the edge length parameter $L_e$ and the restraint parameter $r$. Moreover, we provide the indecomposable direct summands for each persistence module.