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 $π_{n-3}$ for all $n$. We also explore the persistence of these nontrivial classes when the ambient disc is deformed into an ellipse.

Figures (11)

• Figure 1: A configuration $x$ of points in $D^2$ (black) with its stress graph (grey). Each point of $x$ is contained in a circle (grey, dashed), where the radius of these discs is the greatest achievable without any discs overlapping each other or leaving the unit disc. At each point of contact between discs, or between a disc and $\partial D^2$, an edge joins the corresponding vertices.
• Figure 2: The critical configurations and associated critical radii of $\mathrm{Conf}_n$, $n\in \{1, 2,3,4,5 \}$, with the underlying stress graphs and associated configurations of discs. In most pairs $(n,r)$ shown, the critical configuration is unique up to rotation of the unit disc. The exceptions are $\left(5, \frac{1}{4} \right)$ and $\left(5, \frac{1}{3}\right)$, where the isolated disc may move freely; and $\left(4, \frac{1}{3}\right)$, where the outer discs may move freely, provided that the centres are never contained in an open semicircle centred at the origin.
• Figure 3: The angles $\theta_k$ and $\phi_k$ used to construct the map $\mathrm{ang}$ in Def. \ref{['defn: ang']}. Angles are taken to be anticlockwise from the dotted line to the solid line, so $\theta_{k-1}<0$ and $\theta_k, \phi_k>0$ in this diagram.
• Figure 4: A loop of configurations of four discs of radius 0.29 inside the unit disc, starting at the top right, which is homotopic to the sequence $\sigma_3 \sigma_1^{-1} \sigma_3^{-1} \sigma_1$ in the standard presentation of the braid group. This descends to the loop $\partial \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]^2 \subset T^2$ (blue) under the map $\mathrm{ang}$. The blue loop is non-contractible in $T^2 \backslash \{0\}$; then since $0\notin \mathrm{ang}(\mathrm{Conf}_{4, 0.29})$, the loop of configurations is also non-contractible. This homotopy class persists for $\frac{1}{4}< r \le \frac{1}{3}$.
• Figure 5: The placement of the $i$-th disc in the construction of the configurations of Theorem \ref{['thm: nts']}.

Theorems & Definitions (72)

• Definition 1.1.1
• Definition 1.1.2
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Lemma 2.0.1
• Proposition 2.0.2
• proof
• Proposition 2.0.3