Persistent Homology of Configuration Spaces of Trees

TL;DR

This paper investigates the persistent homology of configuration spaces of trees, specifically star and H-graphs, using multiparameter persistence modules to understand the topological structure of robot configurations.

Contribution

It introduces a detailed analysis of 2-parameter persistence modules for configuration spaces of specific trees, including their indecomposable summands, advancing topological robotics understanding.

Findings

Explicit description of 2-parameter persistence modules for star and H-graphs

Identification of indecomposable summands for these modules

Enhanced understanding of robot configuration spaces in topological data analysis

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 ($\mathsf{Star}_k$, $k\geq 3$) 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.