Homeomorphism and Homotopy Types of Restricted Configuration Spaces of Metric Graphs

TL;DR

This paper investigates the topological properties of restricted configuration spaces on finite metric graphs, analyzing how these spaces change with parameters and providing bounds on their isotopy types.

Contribution

It offers a detailed study of the homotopy, homeomorphism, and isotopy types of restricted configuration spaces on metric graphs, including a polynomial bound on isotopy types.

Findings

Homotopy and homeomorphism types depend on the parameter r.

Polynomial upper bound on the number of isotopy types.

Characterization of configuration spaces in topological robotics.

Abstract

For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy, homeomorphism, and isotopy types of these spaces over the space of parameters r and provide a polynomial upper bound (in the number of edges of Gamma) for the number of isotopy types.