Topological complexity of n points on a tree
Steven Scheirer

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