Topological complexity of n points on a tree

TL;DR

This paper calculates the topological complexity of configuration spaces of distinguishable and indistinguishable points on a tree, providing explicit values for many cases, which aids in understanding motion planning in such spaces.

Contribution

It determines the topological complexity of configuration spaces of points on a tree, including both distinguishable and indistinguishable cases, for a wide range of point counts.

Findings

Calculated $TC(UC^n( ext{Gamma}))$ for any tree and many $n$

Derived $TC(C^n( ext{Gamma}))$ from $TC(UC^n( ext{Gamma}))$ for path-connected spaces

Provides explicit topological complexity values for configuration spaces on trees

Abstract

The topological complexity of a path-connected space $X,$ denoted $TC(X),$ can be thought of as the minimum number of continuous rules needed to describe how to move from one point in $X$ to another. The space $X$ is often interpreted as a configuration space in some real-life context. Here, we consider the case where $X$ is the space of configurations of $n$ points on a tree $\Gamma.$ We will be interested in two such configuration spaces. In the, first, denoted $C^n(\Gamma),$ the points are distinguishable, while in the second, $UC^n(\Gamma),$ the points are indistinguishable. We determine $TC(UC^n(\Gamma))$ for any tree $\Gamma$ and many values of $n,$ and consequently determine $TC(C^n(\Gamma))$ for the same values of $n$ (provided the configuration spaces are path-connected).