A traditional configuration space of a metric graph X models n particles existing in a network of tracks with no collisions allowed. If instead of n particles, there are n “robots,” then the resulting space’s type depends on the robots’ sizes and other distance restrictions given by a restraint parameter r. In this talk, we discuss the homotopy and homeomorphism types of these restricted configuration spaces ${X^n_r}$ over the domain of the parameter r and provide polynomial upper bounds (in the number of edges of the graph X) for the number of types.

Based on joint work with Murad Özaydın:

“Homeomorphism Types of Restricted Configuration Spaces of Metric Graphs,” International Mathematics Research Notices, Volume 2018, Issue 20, October 2018, Pages 6329–6348, https://doi.org/10.1093/imrn/rnx061