Thick braids and other non-trivial homotopy in configuration spaces of hard discs

Abstract

We study ordered configuration spaces of $n$ hard discs inside a unit disc, and how the topology changes with the radius $r$ of the hard discs. We describe the full homotopy type of this space for all radii when $n=4$ and exhibit nontrivial classes in ${\pi }_{n-3}$ for all $n$. We also explore the persistence of these nontrivial classes when the ambient disc is deformed into an ellipse.

42 pages, 11 figures. v3: clearer introduction and motivation; new results added and some more speculative sections removed. v2: improved narrative flow, signposting and motivation; improved clarity of some proofs; weakened hypotheses in Thm. 4.1; fixed some typos and improved consistency of notation