| الوصف: |
We investigate the homotopy groups of the configuration space of hard discs inside a unit disc, just beyond the first critical radius. In the four-disc case, we find a non-contractible `thick braid' (loop in the configuration space), which contracts in the ordered configuration space of points when we replace each disc by its centre. For $n\ge 5$ discs, we find a non-contractible $(n-3)$-sphere, and we explore the persistence of this homotopy class when the ambient unit disc is deformed. For sufficiently large $n$, we demonstrate the existence of non-trivial $\pi_k$-classes with $kComment: 21 pages, 6 figures. 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 |
