Persistent Homology of Configuration Spaces of Trees

Abstract

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 (${\mathsf{Star}}_k$, $k\ge 3$) and the generalized H-graph (${\mathcal{H}}_{m,n}$, $m,n\ge 3$) with the edge length parameter $L_e$ and the restraint parameter $r$. Moreover, we provide the indecomposable direct summands for each persistence module.

80 pages