Thick braids and other non-trivial homotopy in configuration spaces of hard discs
November 28, 2023
Abstract
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 \(k<n-3\) beyond higher critical radii.
1 Introduction↩︎
Let \(\mathrm{Conf}_{n,r}(U)\) be the ordered configuration space of \(n\) non-overlapping open discs of radius \(r\) inside some (usually connected) \(U\subset \mathbb{R}^2\). By discs, we mean balls in \(\mathbb{R}^2\) under the standard Euclidean metric. For much of this paper, we focus on the unit disc \(D^2 = B(0,1)\); in this case we suppress the name of the space, simply writing \(\mathrm{Conf}_{n,r}\).
This generalises the ordered configuration space of points, which we denote here as \(\mathrm{Conf}_n(U)\). It is well-known that, when \(U\subset\mathbb{R}^2\) is a connected, simply connected, codimension zero submanifold, the fundamental group of \(\mathrm{Conf}_n(U))\) is the pure braid group on \(n\) strings, and the higher homotopy groups are trivial. However, when the points are replaced with discs, the configuration space develops holes and singularities.
More precisely, for a fixed number \(n\) of discs, it has been shown by Baryshnikov, Bubenik, and Kahle [1] that there is a discrete set of ‘critical radii’ at which the homotopy-type of \(\mathrm{Conf}_{n,r}(U)\) changes as \(r\) increases. They demonstrate this by observing that \(\mathrm{Conf}_{n,r}(U)\) is the superlevel set of a ‘min-type Morse function,’ \(\tau \colon \mathrm{Conf}_n(U) \to \mathbb{R}\), which is defined by the rule \[\tau(x_1, \ldots, x_n) = \min\left\{ \min\left\{\frac{1}{2}|x_i-x_j| \mid i\neq j\right\}, \min\left\{|x_i-p| \mid 1\le i \le n, p\in \partial U\right\} \right\}\] Equivalently, \(\tau\) maps each configuration \((x_1, x_2, \ldots, x_n) \in \mathrm{Conf}_n(U)\) to the greatest radius \(r\) such that replacement of each \(x_i\) by the disc \(B(x_i,r)\) yields an element of \(\mathrm{Conf}_{n,r}(U)\) (i.e. the discs \(B(x_i,r)\) are non-overlapping and lie within \(D^2\)).
Baryshnikov et al. further prove [1] that these critical radii correspond exactly to ‘balanced configurations’ inside \(\mathrm{Conf}_n(U)\), which are characterised by certain weighted graphs whose vertices lie at the centre of each disc and at each point of intersection between a disc boundary and \(\partial U\) (see Figure 1 for the \(n=4\) case).
![Figure 1: The balanced configurations of \mathrm{Conf}_4 with their associated stress graph [1] and critical radius. The critical radius is such that the discs of this radius (shown in grey) touch if their centres are connected by an edge, and touch the boundary if their centre is adjacent to a boundary vertex.](_pdflatex/htyujzxb.png "fig:")
Then in general, \(\mathrm{Conf}_{n,r}(U)\) is homotopy equivalent to \(\mathrm{Conf}_n(U)\) for \(r\) below the least critical radius, and \(\mathrm{Conf}_{n,r}(U)\) is empty beyond the greatest critical radius. Thus the problem of finding the greatest critical radius is a circle packing problem. For \(U=D^2\), Alpert [2] shows that the least critical radius is \(\frac{1}{n}\), and the least critical radius which does not take the form \(\frac{1}{n-k}\) (corresponding to \(n-k\) discs lying on a diameter) is strictly greater than \(\frac{3}{2n+3}\). The greatest critical radius is currently known only for \(1\le n \le 13\) and \(n=19\) (see [3] for a comprehensive summary).
Consider the inclusion \(i_{n,r}\colon \mathrm{Conf}_{n,r}\to\mathrm{Conf}_n\), which maps a configuration of discs to the configuration of their centres. Alpert shows that for \(r\) just above the first critical radius \(\frac{1}{n}\), \(\ker i_{n,r}^*\) is non-empty, where \(i_{n,r}^*\) denotes the pullback to cohomology [2]. For \(n=4\), \(\ker i_{4,r}^*\) increases in size at each critical radius as \(r\) increases [2].
Alpert’s work shows that the homotopy type of \(\mathrm{Conf}_{n,r}\) changes at these critical radii. Since \(\mathrm{Conf}_n\) is the Eilenberg-Maclane space \(K(PBr_n, 1)\), where \(PBr_n\) is the pure braid group on \(n\) strings, it must be the case that beyond these critical radii, either the fundamental group is no longer \(PBr_n\) or one of the higher homotopy groups is non-trivial. In this paper, we observe the following features of the homotopy groups of \(\mathrm{Conf}_{n,r}\) beyond the first few critical radii.
Theorem 1. Let \(\frac{1}{4}<r\le \frac{1}{3}\). Then there is some nonzero \(\gamma \in \pi_1(\mathrm{Conf}_{4,r})\) such that \((i_{n,r})_\ast (\gamma) = 0\).
We may think of this as a ‘thick braid’ beyond the first critical radius: a non-trivial braid of thick strings which becomes trivial when the strings are made thinner. Then the fundamental group of \(\mathrm{Conf}_{4,r}\) for this range of \(r\) is not \(PBr_4\). The next result generalises Theorem 1, proving that for \(n\ge5\) and \(\frac{1}{n}<r \le \frac{1}{n-1}\), \(\mathrm{Conf}_{n,r}\) is not aspherical.
Theorem 2. Take \(n\ge 5\), and let \(\frac{1}{n} < r \le \frac{1}{n-1}\). Then there is some non-contractible \((n-3)\)-sphere in \(\mathrm{Conf}_{n,r}\).
Noting that \(D^2\) is highly symmetric, we next consider the persistence of these homotopy sphere classes under small deformations of the ambient unit disc, as well as larger deformations in the case that \(U\) is an ellipse.
Theorem 3. Take \(r\in (0,\infty)\), \(n\ge 4\), and \(U\subset \mathbb{R}^2\). Suppose there are discs \(C_1, C_2\subset\mathbb{R}^2\) of radius \(\left(n-1\right)r\) and strictly less than \(nr\) respectively, such that \(C_1\subset U\) and any disc of radius \(r\) in \(U\) is contained in \(C_2\). Then there is some non-zero \(\gamma \in \pi_{n-3}(\mathrm{Conf}_{n,r}(U))\) such that \((i_{n,r})_\ast(\gamma)=0 \in \pi_{n-3}(\mathrm{Conf}_n(U))\).
Theorem 4. Let \(n\ge 4\), and \(E\) be the ellipse with semi-major radius \(a\), semi-minor radius \(b\) and eccentricity \(e=\sqrt{1-\frac{b^2}{a^2}}\). Then \(\mathrm{Conf}_{n,r}(E)\) contains a non-contractible \((n-3)\)-sphere of the same construction as that in Thm. 2 for \(r=\frac{1}{n}+\varepsilon\), \(\varepsilon>0\) sufficiently small, when either:
\(b\in \left(\frac{1}{\sqrt{n}}, 1\right]\) and \(a=1\), or
\(b\in \left(\frac{1}{n}, \sqrt{r}\right]\) and \(e^2 = \frac{(1-r)^2}{b^2-r^2+(1-r)^2}\).
In the final section, we explore further critical radii for large \(n\) by choosing \(k<n\) and trying to add \(n-k\) discs to the known \((k-3)\)-sphere in \(\mathrm{Conf}_{k,r}\) for \(r=\frac{1}{k}+\varepsilon\). We make an initial exploration into the question of how you maximise \(n\) for a given value of \(k\), treating this as a parametric packing problem.
Previous work on configuration spaces of objects with volume includes generalisations of the kissing sphere problem [4], [5], discs in a rectangle or infinite strip [6]–[11], and non-rotating squares in a rectangle [12]–[14].
We note, for comparison with Thm. 2, that Baryshnikov et al. [1] demonstrate the existence of a non-trivial \((n-2)\)-sphere in the configuration space of discs in a rectangle, with radius just beyond the first critical radius. This is of similar construction to the homotopy class in Thm. 2, but one dimension higher - it can be seen from the proof that this discrepancy in dimension results from the symmetry of our ambient unit disc.
1.1 Further notation↩︎
In this paper, we denote the configurations of \(\mathrm{Conf}_{n,r}(U)\) by \(\mathcal{D}= (D_1, \ldots, D_n)\), and we denote the centre of each \(D_i\) by \(x_i\) – that is, \(D_i = B(x_i,r)\) for all \(i \in \{1,\ldots,n\}\). We write \(\bigcup \mathcal{D}\mathrel{\vcenter{:}}= \bigcup_{i=1}^n D_i\).
We now define the following angles (see Figure 2), used originally in Arnol’d’s representation of \(H^\ast(\mathrm{Conf}_n(\mathbb{R}^2))\) [15] and subsequently by Alpert [2] to prove the results in cohomology stated above. Let \(\theta_i\) (\(i\in \{1,\ldots,n-1\}\)) be the anticlockwise angle from the horizontal rightward ray through \(x_i\) to the line segment \(\overline{x_ix_{i+1}}\), then set \(\phi_1 = \theta_1\), \(\phi_i = \theta_i - \theta_{i-1}\) for all other \(i\). The angle map is defined to be the continuous function \[\mathrm{ang}_U \colon \mathrm{Conf}_{n,r}(U) \to T^{n-2} \;, \quad \qquad \mathcal{D}\mapsto (\phi_2, \phi_3, \ldots, \phi_{n-1})\] where the name of the space, \(U\), is often suppressed. We may sometimes instead take the codomain to be \(T^k\) for some \(k<n-2\), and truncate the image to \(\mathrm{ang}(\mathcal{D}) = (\phi_2, \ldots, \phi_{k+1})\).
1.2 Acknowledgements↩︎
I owe gratitude to my supervisor, Dr Jonny Evans, for introducing me to this problem and for his regular feedback throughout the process of this research. I am grateful also to Dr Hannah Alpert, who gave positive feedback on an early draft of this paper and gave me an overview of the existing research and researcher networks in this field. Thank you to Dr Tony Nixon for his interest and support. My PhD position is funded by the Engineering and Physical Sciences Research Council.
2 Four discs, and braids of thick strings↩︎
For \(n=4\), the situation is comparatively simple, as there are only three critical radii (see Figure 1). For \(r>\sqrt{2}-1\), the configuration space is empty. For \(\frac{1}{3}<r\le \sqrt{2}-1\), the configuration space is homotopy equivalent to \(\mathrm{Conf}_{4,\sqrt{2}-1}\), in which there is only one configuration up to rotation of the unit disc and permutation of the four discs, so the configuration space is a disjoint union of six circles. For \(r\le \frac{1}{4}\), it is the configuration space of points. Thus, it only remains to check \(\frac{1}{4}<r\le \frac{1}{3}\). In this case we demonstrate the existence of a ’thick braid": a non-trivial loop in \(\mathrm{Conf}_{4,r}\) which is trivial in \(\mathrm{Conf}_4\) when the discs of each configuration are replaced by their centres. Thus it is immediately clear that the thickness of the discs affects the homotopy type of the space, which ceases to be a \(K(PBr_4,1)\) beyond the radius \(\frac{1}{4}\).
Theorem 1. (This is Theorem 1). Let \(\frac{1}{4}< r \le \frac{1}{3}\). Then \(\mathrm{Conf}_{4, r}\) contains a non-trivial loop which is homotopic to \(\sigma_1^{-1} \sigma_3^{-1} \sigma_1 \sigma_3\) in the standard representation of the braid group, and therefore trivial when the disc radii are reduced below \(\frac{1}{4}\).
A realisation of this loop and its image under \(\mathrm{ang}\) in \(T^2 \backslash \{0\}\) is depicted in Figure 3.
Before proving this, we must introduce a lemma which will be required throughout the paper.
Lemma 2. Let \(n,k\in \mathbb{N}\), \(r>\frac{1}{k}\) and \(U\subset \mathbb{R}^2\) a connected, codimension zero submanifold such that any disc of radius \(r\) in \(U\) is contained in \(D^2\). Let \(T\) be some \(m\)-dimensional manifold, \(f\colon \mathrm{Conf}_{n,r}(U) \to T\) be a continuous map, and \(p\in T\) be such that if \(f(\mathcal{D})=p\), then some \(k\) discs in \(\mathcal{D}\) lie on a straight line. Let \(S\) be an \((m-1)\)-sphere in \(T\backslash \{p\}\) such that \([S] \neq 0 \in \pi_{m-1}(T\backslash \{p\})\) and \(\mathcal{S}\colon S\to \mathrm{Conf}_{n,r}(U)\) a local section of \(f\). Then \(\mathcal{S}\) represents a non-trivial class in \(\pi_{m-1}(\mathrm{Conf}_{n,r}(U))\).
Proof. First, we note that \(\mathcal{S}\) is an \((m-1)\)-sphere inside \(\mathrm{Conf}_{n,r}(U)\), so represents some class in \(\pi_{m-1}(\mathrm{Conf}_{n,r}(U))\). Since \(r>\frac{1}{k}\), we cannot fit \(k\) discs of radius \(r\) on one straight line inside \(D^2\), nor therefore inside \(U\), so \(f(\mathrm{Conf}_{n,r}(U)) \subset T\backslash \{p\}\). Thus \(S = f(\mathcal{S})\) is non-trivial in \(\pi_{m-1}(f(\mathrm{Conf}_{n,r}(U)))\), and the result follows. ◻
Proof of Theorem 1. Consider the loop \(S = \partial \left( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]^2 \right) \subset T^2\) . We can then define a local section \(\mathcal{S} \colon S \to \mathrm{Conf}_{4,r}\) for any \(r\le \frac{1}{2+\sqrt{2}}\) by \[(\phi_2, \phi_3) \mapsto \frac{1}{2+\sqrt{2}} \left( \begin{array}{l} (-\sin\phi_2+\sin\phi_3, -1-\cos\phi_2-\cos\phi_3) \\ (\sin\phi_2+\sin\phi_3, -1+\cos\phi_2-\cos\phi_3) \\ (\sin\phi_2+\sin\phi_3, 1+\cos\phi_2-\cos\phi_3) \\ (\sin\phi_2-\sin\phi_3, 1+\cos\phi_2+\cos\phi_3) \end{array} \right)\] where the rows are the coordinates of the centres, denoted \(x_i\) (\(i\in \{1,2,3,4\}\)), of the four discs.
It is possible from the given coordinates to check \(\mathcal{S}\) is well-defined. First note that on \(S\), there is always one \(\phi_j = \pm \frac{\pi}{2}\). Then \(\sin\phi_j = \pm 1\) and \(\cos\phi_j=0\), so letting \(\phi\) represent the other coordinate, \(|x_i|^2 = \left(\frac{1}{2+\sqrt{2}}\right)^2 \left((1\pm \sin\phi)^2 + (1\pm \cos\phi)^2\right) \le \left(\frac{1+\sqrt{2}}{2+\sqrt{2}}\right)^2 \le (1-r)^2\). Therefore, \(D_i\subset D^2\) for each \(i\). We may check similarly that any two disc centres are separated by a distance at least \(2r\).
By construction, discs 2 and 3 lie on the same vertical line at all values of \(\mathcal{S}\), while the coordinates of discs 1 and 4 are chosen to satisfy \(\mathrm{ang}\circ\mathcal{S}= \mathrm{id}_S\), so this is indeed a local section to \(\mathrm{ang}\).
We can also check from the coordinates that this represents \(\sigma_1^{-1} \sigma_3^{-1} \sigma_1 \sigma_3\) as claimed (Fig. 3).
If \(\mathrm{ang}(\mathcal{D}) = 0\), then the four discs lie on a straight line. Furthermore, \([S] \neq 0 \in \pi_{n-3}(T^2\backslash \{0\})\). Therefore, when \(r>\frac{1}{4}\), we have \(4r>1\) and so \(\mathcal{S}\) represents a non-trivial class in \(\pi_1(\mathrm{Conf}_{n,r})\) by Lemma 2.
Finally, this homotopy class persists until the second critical radius [1], which is \(r=\frac{1}{3}\) (see Fig. 1). ◻
![Figure 3: A loop of configurations of four discs of radius 0.29 inside the unit disc, which is homotopic to the sequence \sigma_1^{-1} \sigma_3^{-1} \sigma_1 \sigma_3 in the standard representation of the braid group. This loops descends to the blue loop \partial \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]^2 under the map \mathrm{ang}, which is non-contractible in the image of \mathrm{ang}. Thus the original loop of configurations is non-contractible. This homotopy class persists for \frac{1}{4}< r \le \frac{1}{3}.](_pdflatex/ysigdxok.png "fig:")
3 Extension to more discs and higher homotopy groups↩︎
It is well-known [16] that the configuration space of \(n\) points in any codimension-zero subset of the plane (and in particular the open unit disc) is aspherical. Here, we generalise the work of section 2 to demonstrate a non-trivial element of \(\pi_{n-3}(\mathrm{Conf}_{n,\frac{1}{n}+\varepsilon})\), provided \(\varepsilon\) is suitably small.
Theorem 3. (This is Theorem 2). There is a non-trivial sphere of dimension \(n-3\) in \(\mathrm{Conf}_{n,r}\) for all \(n\ge 4\) and \(\frac{1}{n}<r \le \frac{1}{n-1}\).
To prove this theorem, we will construct a sphere of configurations for sufficiently small \(r>\frac{1}{n}\). The following lemma is used to prove that the configurations in the constructed sphere lie inside \(D^2\).
Lemma 4. Let \(\mathcal{D}= (D_1, D_2, \ldots, D_n)\) be a configuration of open unit discs such that \(D_k\) is in contact with \(D_{k+1}\) for all \(1\le k\le n-1\) and \(\phi_l=\pm\xi\) for some \(2\le l\le n-1\), \(0<\xi\le \frac{\pi}{2}\). Then \(\bigcup \mathcal{D}\subset B\left(\frac{x_1+x_n}{2}, n-4\sin^2\frac{\xi}{4}\right)\).
Proof. Let \(z=\frac{x_1+x_n}{2}\). The claim is equivalent to proving \(|z-x_i| \le n - 4\sin^2\frac{\xi}{4} - 1 = n - 3 + 2\cos\frac{\xi}{2}\) for all \(i\). We have two cases:
\(\boldsymbol{i\neq l}\): By the triangle inequality, \(|z-x_i| \le \frac{1}{2}(|x_1-x_i|+|x_i-x_n|)\). Then, assuming (by reversing the labelling if necessary) that \(i>l\), we have
rCl |x_1-x_i| && _k=1^l-2|x_k-x_k+1| + |x_l-1-x_l+1| + _k=l+1^i-1|x_k-x_k+1|
&=& 2(l-2) + 4 + 2(i-1-l)
&=& 2(i-3) + 4
|x_i-x_n| && _k=i^n-1|x_k-x_k+1|
&=& 2(n-i)
Thus \(|z-x_i| \le n-3+2\cos\frac{\xi}{2}\), as required.
\(\boldsymbol{i=l}\): First, consider our configuration placed in the plane with \(x_{l-1}\) and \(x_{l+1}\) equidistant from the origin along the \(x\)-axis. Then (up to reflection) \(x_{l-1} = (-c, 0)\), \(x_{l} = (0, -s)\), \(x_{l+1} = (c, 0)\), where \(c=2\cos\frac{\xi}{2}\) and \(s=2\sin\frac{\xi}{2}\). As in the first case, we have \(|x_1-x_{l-1}| \le \sum_{i=1}^{l-2} |x_i - x_{i+1}| = 2(l-2)\), \(|x_n-x_{l+1}| \le \sum_{i=l+1}^{n-1} |x_i - x_{i+1}| = 2(n-l-1)\), so there are \(A\in [0,l-2]\), \(B\in [0,n-l-1]\), and angles \(\alpha, \beta\) such that \(x_1 = (-c-2A\sin\alpha, 2A\cos\alpha)\) and \(x_n = (c+2B\sin\beta, 2B\cos\beta)\). Then \(z - x_l = (B\sin\beta - A\sin\alpha, A\cos\alpha + B\cos\beta + s)\). Therefore
rCl |z-x_l|^2 &=& (B- A)^2 + (A+ B+ s)^2
&=& B^2^2- 2AB+ A^2^2
&& + B^2^2+ 2AB+ A^2^2+ s^2 + 2sA+ 2sB
&=& B^2 + 2AB(+) + A^2 + s^2 + 2sA+ 2sB
&& B^2 + 2AB + A^2 + s^2 + 2sA + 2sB
&=& (A+B+s)^2
&& (n-3+2)^2
We complete our proof by noting that \(\sin\frac{\xi}{2} \le \cos\frac{\xi}{2}\). ◻
Remark 5. This maximum is sharp, as it is attained by discs \(1\) and \(n\) in the configuration in which all discs except \(l\) are collinear.
Proof of Theorem 3. First, consider the case \(\frac{1}{n} < r \le \frac{1}{n-4\sin^2(\frac{\pi}{2n})}\). We take the \((n-3)\)-sphere \(S=\partial\left[ -\frac{2\pi}{n}, \frac{2\pi}{n} \right]^{n-2}\), and claim that the following algorithm defines a local section \(\mathcal{S}\colon S \to \mathrm{Conf}_{n,r}\). We construct the configuration \((D_1, \ldots, D_n) \mathrel{\vcenter{:}}= \mathcal{S}(\phi_2, \ldots, \phi_{n-1})\) as follows:
Place \(D_1\) centred at the origin of the plane and \(D_2\) in contact with it, centred at \((2r, 0)\).
Place each subsequent disc \(D_i\) in contact with \(D_{i-1}\), such that \(x_i\) lies on the ray from \(x_{i-1}\) at angle \(\phi_{i-1}\).
Translate all discs by \(-\frac{x_1+x_n}{2}\).
\(\mathrm{ang}\circ\mathcal{S}= \mathrm{id}_S\) by construction (step 2). The placement of \(D_i\) at step 2 is continuous with respect to \(\phi_i\), and the translation in step 3 is continuous with respect to \(x_1\) and \(x_n\), so \(\mathcal{S}\) is continuous. Thus, we prove our claim if we show that \(\mathcal{S}(\Phi) \in \mathrm{Conf}_{n,r}\) for all \(\Phi\in S\).
At the placement of \(D_i\) in step 2, consider the two regular \(n\)-gons which have \(\overline{x_{i-1}x_i}\) as one of their sides. Since \(|\phi_k|\le \frac{2\pi}{n}\), the external angle of a regular \(n\)-gon, for all \(k\), the sequence of edges \[x_1 \longrightarrow x_2 \longrightarrow \ldots \longrightarrow x_{i-1} \longrightarrow x_i\] lies outside or on the boundary of these two \(n\)-gons. Thus, since \(i-1<n\), \(D_i\cap D_j = \emptyset\) for \(j\in\{1,2,\ldots,i-1\}\). Therefore no two discs in \(\mathcal{S}(\phi_2, \ldots, \phi_{n-1})\) overlap. Since every point in \(S\) has at least one coordinate equal to \(\pm \frac{2\pi}{n}\) and \(D_{i+1}\) touches \(D_i\) for all \(i\), we can apply Lemma 4 with \(\xi=\frac{2\pi}{n}\) to show that \(\bigcup \mathcal{S}(\phi_2, \ldots, \phi_{n-1}) \subset B\left(0, \left(n-4\sin^2\frac{\pi}{2n}\right)r\right) \subset D^2\), which completes the proof of our claim.
Next, we note that if \(\mathrm{ang}(\mathcal{D})=0\), then the \(n\) discs lie on one straight line. Moreover, \([S] \neq 0 \in \pi_{n-3}(T^{n-2}\backslash \{0\})\). Therefore we may apply Lemma 2 to show that \(\mathcal{S}\) represents a non-trivial class in \(\pi_{n-3}(\mathrm{Conf}_{n,r})\).
Finally, this non-trivial class persists up to the second critical radius [1]. This is \(\frac{1}{n-1}\) – we may show this by hand for \(n\in \{4,5\}\), while for \(n\ge 6\), we note that \(\frac{3}{2n+3} \ge \frac{1}{n-1}\), and so this must be the second critical radius [2]. ◻
Remark 6. While we are able to extend the existence of a homotopic sphere in \(\mathrm{Conf}_{n,r}\) to radii up to \(r=\frac{1}{n-1}\) in the proof of Thm. 3, Remark 5 shows this will not project to \(\partial \left( [-\frac{2\pi}{n}, \frac{2\pi}{n}]^{n-2}\right)\) for radii above \(\frac{1}{n-4\sin^2(\frac{\pi}{2n})}\): for these radii, any configuration with angles \((0, \ldots, 0, -\frac{\pi}{n}, \frac{2\pi}{n}, -\frac{\pi}{n}, 0, \ldots, 0)\) has diameter greater than 2, so does not lie in the unit disc.
Remark 7. This algorithm allows us to construct explicit coordinates for our configuration in terms of \(\theta_i = \sum_{j=1}^i \phi_j\). At step 2, \(x_i = 2r(\sum_{j=1}^{i-1}
\cos\theta_j, \sum_{j=1}^{i-1} \sin\theta_j)\) for
\(i\ge 2\), with \(\theta_1 = 0\). After translation by \(-\frac{1}{2}(x_1+x_n) = -r(\sum_{j=1}^{n-1} \cos\theta_j, \sum_{j=1}^{n-1} \sin\theta_j)\), we have
\(x_i = r (\sum_{j=1}^{i-1} \cos\theta_j - \sum_{j=i}^{n-1} \cos\theta_j, \sum_{j=1}^{i-1} \sin\theta_j - \sum_{j=i}^{n-1} \sin\theta_j)\).
Remark 8. We can construct a sphere of the same homotopy class by lifting \(S=\partial \left( [-\xi, \xi]^{n-2}\right)\) to \(\mathrm{Conf}_{n,r}\) by the same algorithm for all \(0<\xi\le \frac{2\pi}{n}\) and \(\frac{1}{n} < r \le \frac{1}{n-4\sin^2(\frac{\xi}{4})}\). As \(\xi\) decreases, the variation of the configuration about the line through \(x_1\) and \(x_2\) will decrease - that is, for each disc \(D_i\), the maximal distance from \(x_i\) to this line over all configurations will decrease.
Remark 9. This construction gives rise to many distinct homotopy classes in \(\pi_{n-3}(\mathrm{Conf}_{n,r})\) for \(\frac{1}{n}< r \le \frac{1}{n-1}\). Given \(r \le \frac{1}{n-4\sin^2 \frac{\pi}{2n}}\), take \(\sigma\in S_n\), and let \(\mathcal{S}'=\sigma\cdot\mathcal{S}\), where the action of \(\sigma\) on \(\mathrm{Conf}_{n,r}\) permutes the discs in the natural way. It can be shown that there is a path \(\gamma\colon [0,1]\to \mathrm{Conf}_{n,r}\) such that \(\gamma(0) \in \mathcal{S}\) and \(\gamma(1) = \sigma\cdot \gamma(0)\), so we can consider \([\mathcal{S}']\) in the same homotopy group as \([\mathcal{S}]\). Then \(\mathrm{ang}(\mathcal{S}')\) encircles some \(0\neq \Phi \in \{(\phi_2, \ldots, \phi_{n-1})\in T^{n-2} \mid \forall_i \;\phi_i\in \{0,\pi\}\}\), where \(\Phi\) corresponds to \(n\) discs lying on a line. Hence \(\mathrm{ang}_*([\mathcal{S}']) \neq [S]\), so \([\mathcal{S}'] \neq [\mathcal{S}]\).
4 Persistence under deformation of the unit disc↩︎
Given the rotational symmetry of \(D^2\), \(\tau\) behaves as a Morse-Bott function, with each critical radius corresponding not to a single critical configuration, but to an \(S^1\) of critical configurations. Under small perturbations, we would thus expect, generically, a critical level \(r_0\) to split into two critical radii, say \(r_-<r_+\) near \(r_0\). Provided the perturbation is small, we would expect the topology just above \(r_+\) in the perturbed case to be the same as the topology just above \(r_0\) in the unperturbed case. By contrast, the homotopy type on the region with radius just above \(r_-\) may have different topology, and this is where the \(\pi_{n-2}\)-class of Baryshnikov et al. [1] lives (if you consider an infinite strip as the limit of an ellipse with fixed semi-minor radius as eccentricity approaches 1).
In this section, we investigate the topology beyond \(r_+\), by considering the persistence of our homotopy class from §3 under perturbations of \(D^2\). First, we consider arbitrary small perturbations. Then, following Remark 8, we consider to what extent it is necessary to use the full unit disc by investigating the persistence of this homotopy class as we replace \(D^2\) with an ellipse of decreasing semi-minor radius.
Theorem 10. (This is Theorem 3). Take \(r\in (0,\infty)\), \(n\ge 4\), and \(U\subset \mathbb{R}^2\). Suppose there are discs \(C_1, C_2\subset\mathbb{R}^2\) of radius \(\left(n-1\right)r\) and strictly less than \(nr\) respectively, such that \(C_1\subset U\) and any disc of radius \(r\) in \(U\) is contained in \(C_2\). Then there is some non-zero \(\gamma \in \pi_{n-3}(\mathrm{Conf}_{n,r}(U))\) such that \((i_{n,r})_\ast(\gamma)=0\) in \(\pi_{n-3}(\mathrm{Conf}_n(U))\).
In particular, this shows that the non-trivial \(\pi_{n-3}\)-class remains in \(\mathrm{Conf}_{n,r}(U)\) for \(\frac{1}{n} < r < \frac{1}{n-1}\) when \(U\) is a deformation of \(D^2\), provided that the deformation is small enough that: 1) we can fit a disc of radius \(\left(n-1\right)r\) inside \(U\), and 2) every disc of radius \(r\) in \(U\) lies in the unit disc. The first condition means the boundary of \(U\) cannot move far towards the origin. On the other hand, these conditions place no restrictions on the area of \(U\), nor do they require \(U\) to be bounded – consider, for example, the deformation of \(D^2\) in which we remove the segment of \(D^2\) with \(y\)-coordinate greater than \(1-\varepsilon\) for some small \(\varepsilon>0\), and then take the union of the remaining region with the semi-infinite strip \([-\varepsilon, \varepsilon] \times [0,\infty)\).
Proof. The conditions on \(C_1, C_2\) yield \(\mathrm{Conf}_{n,r}(C_1) \subset \mathrm{Conf}_{n,r}(U) \subset \mathrm{Conf}_{n,r}(C_2)\). Denote the inclusion maps by \(\iota, \iota'\) respectively. Furthermore, it is impossible for \(n\) discs of radius \(r\) to lie on a straight line inside \(C_2\), so \(\mathrm{ang}(\mathrm{Conf}_{n,r}(C_2)) \subset T^{n-2}\backslash\{0\}\).
By Thm. 1 and 3, there is some non-contractible \(\mathcal{S}\in \mathrm{Conf}_{n,r}(C_1)\) such that \((i_{n,r})_\ast([\mathcal{S}])=0\) in \(\pi_{n-3}(\mathrm{Conf}_n(C_1))\). In particular, this is proven by showing that \(\mathrm{ang}(\mathcal{S})\) is non-contractible in \(T^{n-2}\backslash \{0\}\). We claim \(\gamma \mathrel{\vcenter{:}}= \iota_\ast([\mathcal{S}])\) satisfies the desired properties.
For the first property, note that \(\mathrm{ang}_{C_1} = \mathrm{ang}_{C_2} \circ \iota' \circ \iota\). Since \((\mathrm{ang}_{C_1})_\ast([\mathcal{S}]) \neq 0\) in \(\pi_{n-3}(T^{n-2} \backslash \{0\})\), it follows that \(\gamma \neq 0\) in \(\pi_{n-3}(\mathrm{Conf}_{n,r}(U))\).
For the second property, let \(\iota'' \colon \mathrm{Conf}_n(C_1) \hookrightarrow \mathrm{Conf}_n(U)\) be the inclusion map. Then \(i_{n,r} \circ \iota = \iota'' \circ i_{n,r}\) and therefore \[(i_{n,r})_\ast (\gamma) = (i_{n,r})_\ast (\iota_\ast ([\mathcal{S}])) = \iota''_\ast ((i_{n,r})_\ast ([\mathcal{S}])) = \iota''_\ast(0) = 0\] ◻
Following Remark 8, it is apparent we shouldn’t need the full extent of the unit disc in Thm. 3. Indeed, if a small enough angle is used in place of \(\frac{2\pi}{n}\) in the given construction of the \((n-3)\)-sphere, the discs should remain within some narrow strip parallel to (and centred on) the \(x\)-axis. The following result shows that this homotopy class in §3 remains when we deform the unit disc into an ellipse with semi-major radius 1, provided that the semi-minor radius is greater than \(\frac{1}{\sqrt{n}}\). This result also enables us to generalise Thm. 10 by replacing \(C_1\) with a suitable ellipse.
Under this deformation, the first critical level \(\frac{1}{n}\) splits into two critical levels. The lower of these is \(\frac{\sqrt{1-e^2}}{n}\), where the numerator is the semi-minor radius. This theorem concerns the upper critical level, \(\frac{1}{n}\), which is related to the semi-major radius.
Theorem 11. (This is part (i) of Theorem 4). Let \(n\in \mathbb{N}\), and let \(E\) be an ellipse of semi-major radius 1 and eccentricity \(e\in \left[0,\sqrt{\frac{n-1}{n}}\right)\) (equivalently, the semi-minor radius of \(E\) lies in \(\left(\frac{1}{\sqrt{n}}, 1\right]\)). Then there is some \(\varepsilon>0\) such that, for \(r\in\left(\frac{1}{n}, \frac{1}{n}+\varepsilon\right]\), \(\mathrm{Conf}_{n,r}(E)\) contains a non-contractible \((n-3)\)-sphere.
To prove this, we will construct a sphere in \(\mathrm{Conf}_{n,r}(E)\), and prove that the disc centres in each configuration lie in some smaller ellipse \(E'\). The following two lemmas define the necessary relationship between the dimensions of \(E\) and \(E'\) which ensures that, if a disc has its centre in \(E'\), then the disc is contained in \(E\).
Lemma 12. [17] Let \(E\) be an ellipse of semi-major radius \(a\) and eccentricity \(e\) in the plane, centred at the origin, whose major axis lies along the \(x\)-axis, and \(p=(p_1,0)\) a point on the major axis of \(E\). If \(|p_1|>ae^2\), then \(\mathrm{dist}(p,\partial E) = a-|p_1|\), and this is achieved at either \((-a,0)\) or \((a,0)\); otherwise, \(\mathrm{dist}(p,\partial E) = \sqrt{\left(a^2-\frac{p_1^2}{e^2}\right) (1-e^2)}\), and this is achieved at \(\left(\frac{p_1}{e^2}, \pm (1-e^2)\sqrt{\frac{a^2-p_1^2}{e^4}}\right)\).
Lemma 13. Take \(\mathcal{D}\in \mathrm{Conf}_{n,r}(\mathbb{R}^2)\), \(a>r\). Let \(E'\) be an ellipse with semi-major radius \(a-r\) and eccentricity \(e'\) such that the centre of every disc of \(\mathcal{D}\) lies in \(\overline{E'}\). Let \(E\) be a concentric, coaxial ellipse of semi-major radius \(a\) and eccentricity \(e \le \sqrt{\left(1-\frac{r}{a}\right)\left(1-\sqrt{1-(e')^2}\right)}\). Then \(\bigcup \mathcal{D}\subset E\).
Proof. It is sufficient to check this for \(e=\sqrt{\left(1-\frac{r}{a}\right)\left(1-\sqrt{1-(e')^2}\right)}\), since the associated ellipse is contained inside the ellipses of smaller eccentricity. For this proof, we use the equivalent definition \(e=\sqrt{\frac{a-r-b'}{a}}\), where \(b'=(a-r)\sqrt{1-(e')^2}\) is the semi-minor radius of \(E'\). Since \(E\) and \(E'\) are concentric and coaxial, we choose coordinates so that the shared centre is the origin, and the shared major axis is the \(x\)-axis. Take \(q=(q_1,q_2)\in \overline{E'}\) and assume by symmetry that \(q\) lies in the first quadrant. We need to show that \(\mathrm{dist}(q,\partial E)\ge r\). We will achieve this by finding a suitable point on the \(x\)-axis and using the triangle inequality.
Let \(q_3 = \min\{q_1,ae^2\}\) and \(q'=(q_3,0)\). Since \(q_3\le ae^2\), we have \(\mathrm{dist}(q',\partial E) = \sqrt{\left(a^2-\frac{q_3^2}{e^2}\right) (1-e^2)}\) by Lemma 12. The same inequality also yields \(a^2-\frac{q_3^2}{e^2} \ge a^2(1-e^2)\), so that \(\mathrm{dist}(q', \partial E)^2 \ge a^2(1-e^2)^2 = a^2(1-\frac{a-r-b'}{a})^2 = (r+b')^2\).
Then it remains to show \(|q-q'|\le b'\). If \(q_1\le ae^2\), then \(|q-q'|=q_2 \le b'\), where the second inequality arises from the fact that \(b'\) is the semi-minor radius. Otherwise, \(q' = (ae^2,0)\), and \(\sup\{|q-q'| \colon q=(q_1,q_2) \in \overline{E'}, q_1>ae^2\}\) is achieved by some \((q_1,q_2) \in \partial E'\) with \(q_1 \ge ae^2\). Since \(ae^2\le (a-r)(e')^2\), Lemma 12 applied to \(q'\) and \(E'\) shows that \(\partial {E'} \to \mathbb{R}\), \(q \mapsto |q-q'|\) has its local minima at \(\left(\frac{p_1}{e^2}, \pm (1-e^2)\sqrt{\frac{a^2-p_1^2}{e^4}}\right)\), and therefore has local maxima at \((\pm(a-r),0)\). When we restrict to \(q_1 \ge ae^2\), this supremum must be achieved at either: 1) \(q_1=ae^2\), where \(|q-q'|\le b'\) by the first case; or 2) \((a-r,0)\), where \(|q-q'| = a-r-ae^2 = b'\).
Thus \(\mathrm{dist}(q,\partial E) \ge \mathrm{dist}(q',\partial E)-|q-q'| \ge \mathrm{dist}(q',\partial E)-b' \ge r\) as required. ◻
Remark 14. The inequality \(e \le \sqrt{\left(1-\frac{r}{a}\right)\left(1-\sqrt{1-(e')^2}\right)}\) is asymptotically sharp as \(e'\to 1\). To see this, note that when \(e'=1\), we have \(e>\sqrt{1-\frac{r}{a}}\) if and only if \(a-r<ae^2\). Hence, by Lemma 12, \(\mathrm{dist}((a-r,0), \partial E) < |(a,0)-(a-r,0)| = r\), so the disc of radius \(r\) centred at \((a-r,0) \in \overline{E'}\) is not contained in \(E\). This is the case of interest in the following proof.
Proof of Theorem 11. Take some angle \(0<\xi<\frac{\pi}{n-2}\) such that \(1-r>e^2\), where \(r=\frac{1}{n-4\sin^2\frac{\xi}{4}}\), and let \(S = \partial \left( \left[-\xi, \xi \right] \right)^{n-2} \subset T^{n-2}\). Let \(e'=\sqrt{1-\left(1-\frac{e^2}{1-r}\right)^2}\), \(z_1 = (-e'(1-r), 0)\), \(z_2 = (e'(1-r),0)\). Note that \(\frac{e^2}{1-r}< 1\) and \(\frac{e^2}{1-r}\to 1\) as \(e\to\sqrt{\frac{n-1}{n}}^-\), \(\xi\to0\). Thus \(e'<1\) and \(e'\to 1\) as \(e\to\sqrt{\frac{n-1}{n}}^-\), \(\xi\to0\). We construct some \(\mathcal{S}\colon S\to \mathrm{Conf}_{n,r}(\mathbb{R}^2)\) by the following algorithm on each \((\phi_2, \ldots, \phi_{n-1})\in S\):
Place \(D_1\) at the origin of the plane and \(D_2\) in contact with it, centred at \((2r, 0)\).
Place each subsequent disc \(D_i\) for \(i\le n\) in contact with \(D_{i-1}\) such that \(x_i\) lies on the ray from \(x_{i-1}\) at angle \(\phi_{i-1}\) (Figure 4).
Translate and rotate the configuration so that \(x_1\) and \(x_n\) lie on the \(x\)-axis, equidistant from the origin.
This is continuous and the discs are non-overlapping by the arguments in the proof of Theorem \(\ref{thm:32nts}\).
We now show that the centre of every disc in each configuration in \(\mathcal{S}\) fits inside the ellipse with semi-major radius \(1-r\) and foci \(z_1,z_2\), which has eccentricity \(e'\). By remark 7, we see that, at stage 2, \[\begin{align} |x_1-x_n|^2 & = 4r^2 \left| \left( \sum_{i=1}^{n-1}\cos\theta_i \;, \;\sum_{i=1}^{n-1}\sin\theta_i \right) \right|^2 \\ & = 4r^2 \left( \left( \sum_{i=1}^{n-1}\cos\theta_i\right)^2 + \left(\sum_{i=1}^{n-1}\sin\theta_i \right)^2 \right) \\ & = 4r^2 \left( \sum_{i=1}^{n-1} (\cos^2\theta_i + \sin^2\theta_i) + 2\sum_{i=2}^{n-1}\sum_{j=1}^{i-1} (\cos\theta_i \cos\theta_j + \sin\theta_i \sin\theta_j) \right) \\ & = 4r^2 \left( n-1 + 2\sum_{i=2}^{n-1}\sum_{j=1}^{i-1} \cos(\theta_i - \theta_j) \right) \end{align}\] where \(|\theta_i-\theta_j| = |\sum_{k=i+1}^j \phi_k| \le (j-i)\xi \le (n-2)\xi\). Denoting \(\theta = (n-2)\xi < \pi\), we see that \(\cos(\theta_i-\theta_j) \ge \cos\theta\), and thus \[\begin{align} |x_1-x_n|^2 & \ge 4r^2 \left( n-1 + 2\sum_{i=2}^{n-1}\sum_{j=1}^{i-1} \cos\theta \right) \\ & = 4r^2 \left( n-1 + (n-1)(n-2) \cos\theta \right) \\ & = 4r^2 \left( (n-1)^2 - (n-1)(n-2)(1-\cos\theta) \right) \end{align}\] That is, at step 3, the \(x\)-coordinate of \(x_n\) will be at least \[r\sqrt{(n-1)^2 - (n-1)(n-2)(1-\cos\theta)} \to \frac{n-1}{n}\] as \(\xi\to0\). Since \[\begin{align} e'(1-r)& = \sqrt{(1-r)^2-(1-r-e^2)^2} \\ & = \sqrt{2(1-r)e^2-e^4} \\ & < \sqrt{2\left(1-\frac{1}{n}\right)e^2-e^4} \\ & = \sqrt{\left(1-\frac{1}{n}\right)^2-\left(1-\frac{1}{n}-e^2\right)^2} \\ & < 1-\frac{1}{n} \\ & = \frac{n-1}{n} \end{align}\] and \(\sqrt{2\left(1-\frac{1}{n}\right)e^2-e^4}\) is independent of \(\xi\), it follows that \[r\sqrt{(n-1)^2 - (n-1)(n-2)(1-\cos\theta)} > e'(1-r)\] for sufficiently small \(\xi\). Then \(x_1, x_n\) are further from the origin than \(z_1, z_2\) for small \(\xi\), so we can find \(t\in(0,1)\) such that \(z_1 = tx_1 + (1-t)x_n\) and \(z_2 = (1-t)x_1 + tx_n\).
Using this, we find that for any \(i\), \[\begin{align} |x_i-z_1| + |x_i-z_2| & \le t|x_i-x_1| + (1-t)|x_i-x_n| + (1-t)|x_i-x_1| + t|x_i-x_n| \\ & = |x_i-x_1| + |x_i-x_n| \\ & \le 2r\left(n-1-4\sin^2\frac{\xi}{4}\right) \end{align}\] where the last follows from Lemma 4. That is, each \(x_i\) lies inside the closed ellipse with foci \(z_1,z_2\) and semi-major radius \((n-4\sin^2 \frac{\xi}{4})r - r = 1-r\) as required.
Therefore, since \(e = \sqrt{(1-r)\left(1-\sqrt{1-(e')^2}\right)}\), we observe that for all \(\Phi \in S\), \(\bigcup \mathcal{S}(\Phi) \subset E\) by Lemma 13. Thus \(\mathcal{S}\) is a local section of \(\mathrm{ang}\colon \mathrm{Conf}_{n,r}(E) \to T^{n-2}\).
We finish by observing that \(E \subset D^2\), and moreover \([S] \neq 0 \in \pi_{n-3}(T^{n-2}\backslash \{0\})\), where \(0\in T^{n-2}\) corresponds to all discs lying in a straight line. Therefore \(\mathcal{S}\) represents a non-trivial class in \(\mathrm{Conf}_{n,r}(E)\) by Lemma 2. ◻
While \(\frac{1}{\sqrt{n}}\) is the best semi-minor radius we can achieve while keeping the semi-major radius equal to 1, we would hope to achieve better, since, as made clear in Remark 8, the maximal distance of our configurations from the \(y\)-axis is arbitrarily close to \(\frac{1}{n}\). We achieve this semi-minor radius in the next theorem. However, in order to achieve this, we must allow our semi-major radius, and thus too the area of the ellipse, to approach infinity.
Theorem 15. (This is part (ii) of Thm. 4) Let \(n\in \mathbb{N}\), and let \(E\) be an ellipse with eccentricity \(e\in \left[\sqrt{1-r},1\right)\) and semi-minor radius \(b=\sqrt{ \frac{(1-r)^2(1-e^2)}{e^2} + r^2}\), where \(r=\frac{1}{n}+\varepsilon\) for some sufficiently small \(\varepsilon>0\). Then \(\mathrm{Conf}_{n,r}(E)\) contains a non-contractible \((n-3)\)-sphere.
Under these hypotheses, \(b\) takes all values in \(\left(r, \sqrt{r}\right]\). The semi-major radius is \(b\left(1-e^2\right)^{-\frac{1}{2}} = \sqrt{ \frac{(1-r)^2}{e^2} + \frac{r^2}{1-e^2} }\), which takes all values in \(\left[1, \infty\right)\).
Proof. For a sufficiently small angle \(\xi>0\), let \(S=\partial \left[-\xi, \xi\right]^{n-2} \subset T^{n-2}\) and take \(\frac{1}{n}<r \le
\frac{1}{n-4\sin^2\frac{\xi}{4}}\). Take the length-preserving coordinate system in which \(E\) is centred at the origin and its major axis coincides with the \(x\)-axis. We define a
local section \(\mathcal{S}\colon S\to \mathrm{Conf}_{n,r}(E)\) to \(\mathrm{ang}\) by constructing each
\(\mathcal{S}(\phi_2, \ldots, \phi_{n-1})\) as follows:
Place \(D_1\) at the origin of the plane and \(D_2\) in contact with it at \((2r, 0)\).
Place each subsequent disc \(D_i\) in contact with \(D_{i-1}\), such that \(x_i\) lies on the ray from \(x_{i-1}\) at angle \(\phi_{i-1}\).
Translate all discs by \(-\frac{x_1+x_n}{2}\).
Rotate the configuration about the origin so that \(x_1\) and \(x_n\) lie on the \(x\)-axis.
There is no overlap between discs and \(\mathcal{S}\) is continuous by the arguments of Thm. 3, so we simply need to show \(\bigcup \mathcal{S}(\Phi) \subset E\) for all \(\Phi \in S\). By Remark 7, we have at step 3: \[x_i = r \left(\sum_{j=1}^{i-1} \cos\theta_j - \sum_{j=i}^{n-1} \cos\theta_j, \sum_{j=1}^{i-1} \sin\theta_j - \sum_{j=i}^{n-1} \sin\theta_j\right)\] where \(\theta_j = \sum_{l=1}^j \phi_l \to 0\) as \(\xi\to0\). Thus \(x_i \to (\frac{2i-n-1}{n}, 0)\). In particular, \(x_1 \to (-\frac{n-1}{n}, 0)\) and \(x_n\to (\frac{n-1}{n},0)\), so the size of the rotation in step 4 tends to 0, which means this limit position holds after step 4 as well. This limit is the first critical configuration, consisting of the \(n\) discs of radius \(\frac{1}{n}\) lined up along the horizontal diameter of \(D^2\), such that \(D_i\) touches \(D_{i-1}\) and \(D_{i+1}\) if they exist, and \(D_1\) and \(D_n\) touch the boundary.
More precisely, for each \(\frac{1}{n}<r \le \frac{1}{n-4\sin^2\frac{\xi}{4}}\), there is some \(s>r\) (where \(s\to \frac{1}{n}\) as \(\xi\to 0\)) such that, for \(2\le i \le n-1\), \(D_i \subset B\left(\left(\frac{2i-n-1}{n},0\right), s\right)\). In particular, \[\bigcup_{i=2}^{n-1} D_i \subset F_1 \mathrel{\vcenter{:}}= \left[-\frac{n-3}{n}-s, \frac{n-3}{n}+s\right] \times [-s,s]\] We have fixed the centres of \(D_1\) and \(D_n\) to lie on the \(x\)-axis, and Lemma 4 gives us \(|x_1|, |x_n| \le (n-1-4\sin^2\frac{\xi}{4})r \le 1-r\). Thus \[D_1\cup D_n \subset F_2 \mathrel{\vcenter{:}}= B((-1+r,0), r) \cup ([-1+r,1-r] \times [-r,r]) \cup B((1-r,0),r)\] For sufficiently small \(\xi\), we have \(\frac{n-3}{n}+s < 1-r\), and so \[\mathcal{S}(\Phi) \subset F\mathrel{\vcenter{:}}= B((-1+r,0), r) \cup ([-1+r,1-r] \times [-s,s]) \cup B((1-r,0),r) \supset F_1\cup F_2\] for all \(\Phi \in S\) (see Fig. 5).
Next, we show that \(F\subset E\) by showing that all points of \(F\) satisfy the inequality which defines \(E\), \[x^2(1-e^2)+y^2\le b^2 = \frac{(1-r)^2(1-e^2)}{e^2} + r^2\]
First, if \((x,y) \in [-1+r,1-r] \times [-s,s]\), then \(x^2(1-e^2)+y^2 \le (1-r)^2(1-e^2)+s^2\). As \(\xi \to 0\), then \(s,r\to \frac{1}{n}\), and therefore \(s^2-r^2 \to 0\). Since the lower bound on \(e\) is always less than \(\sqrt{\frac{n-1}{n}}\), we may assume \(e\) to be fixed. Therefore \(s^2-r^2 \le \frac{1}{4}(1-e^2)\left(\frac{1}{e^2}-1\right) \le (1-r)^2(1-e^2)\left(\frac{1}{e^2}-1\right)\) for sufficiently small \(\xi\). Then \[\begin{align} x^2(1-e^2)+y^2 & \le (1-r)^2(1-e^2) + r^2 + (1-r)^2(1-e^2)\left(\frac{1}{e^2}-1\right) \\ & = \frac{(1-r)^2(1-e^2)}{e^2} + r^2 \end{align}\]
Second, if \((x,y) \in B((1-r,0),r)\), then \((x-1+r)^2+y^2 \le r^2\). Thus \[\begin{align} x^2(1-e^2)+y^2 & \le x^2(1-e^2) + r^2 - (x-1+r)^2 \\ & = -e^2x^2+2(1-r)x+2r-1 \\ & = -\left(ex-\frac{1-r}{e}\right)^2 + \frac{(1-r)^2}{e^2} + 2r-1 \\ & \le \frac{(1-r)^2}{e^2} + 2r-1 \\ & = \frac{(1-r)^2(1-e^2)}{e^2} + (1-r)^2 + 2r-1 \\ & = \frac{(1-r)^2(1-e^2)}{e^2} + r^2 \end{align}\] Symmetrically, we see the same for \((x,y) \in B((-1+r,0),r)\). Therefore \(\bigcup \mathcal{S}(\Phi) \subset F \subset E\) for all \(\Phi \in S\), which completes the proof that \(\mathcal{S}\) is a local section of \(\mathrm{ang}\).
Finally, let \(\Delta = \{B(p,r) \colon p\in \mathbb{R}^2, \;B(p,r) \subset E\}\) be the set of all discs of radius \(r\) contained in \(E\). We want to show that every such disc also lies in \(D^2\). It is clear geometrically that \(\sup \{|p| \colon B(p,r) \in\Delta \}\) is achieved on the major axis of \(E\), which is the \(x\)-axis. We may assume the \(x\)-coordinate of \(p\) to be positive by symmetry. Then we note \(\partial B((1-r,0),r)\) intersects \(\partial E\) at \(p_{\pm}=\left(\frac{1-r}{e^2}, \pm \sqrt{r^2-(1-r)^2(e^{-2}-1)^2}\right)\), so the \(x\)-coordinate of \(p\) cannot be greater than \(1-r\) – that is, \(\sup \{|p| \colon B(p,r) \in\Delta \} \le 1-r\). This is equivalent to the claim.
Now \([S]\neq 0 \in \pi_{n-3}(T^{n-2}\backslash \{0\})\), and if \(\mathrm{ang}(\mathcal{D}) = 0\), then all the disc centres are collinear. Thus \(\mathcal{S}\) represents a non-trivial element of \(\pi_{n-3}(\mathrm{Conf}_{n,r}(E))\) by Lemma 2, as required. ◻
We note that there is no discontinuity in the ranges of the semi-minor and semi-major radii between Theorems 11 and 15; indeed, when \(e=\sqrt{1-r}\), both settings yield the ellipse of semi-major radius 1 and semi-minor radius \(\sqrt{r}\). Hence there exists a continuous family of ellipses, indexed over the semi-minor radius on the range \(\left(\frac{1}{n}, 1\right]\), which contain this non-contractible sphere class.
5 Higher critical radii: a parametric packing problem↩︎
In this investigation, it is natural to consider whether similar constructions might work beyond higher critical radii. In particular, given a critical radius \(\frac{1}{k}\) (\(k<n\)) for \(\mathrm{Conf}_{n,r}\), corresponding to a configuration with \(k\) discs lying across a diameter, can we find a representative \((k-3)\)-sphere in \(\mathrm{Conf}_{n,\frac{1}{k}+ \varepsilon}\) using \(k\) discs, or does the presence of the remaining \(n-k\) spheres mean this sphere no longer fits? Theorems 11 and 15 show that we choose the sphere so that all configurations remain within an ellipse \(E\); if we can fit \(n-k\) discs in \(D^2 \backslash E\), then our sphere exists in \(\mathrm{Conf}_{n,\frac{1}{k}+ \varepsilon}\) for sufficiently small \(\varepsilon>0\), which gives us a lower bound. In this section, we improve upon this bound, and discuss this as a parametric packing problem: if we fix \(k\), what is the greatest \(n\) for which this homotopy class remains?
Formally, for \(n\ge k\), let \(F_{n,k}\colon \mathrm{Conf}_{n,r}\to\mathrm{Conf}_{k,r}\) be the forgetful map
\((D_1,\ldots,D_n) \mapsto (D_1,\ldots, D_k)\). Then we ask the following.
Question 16. Take \(m,k\in\mathbb{N}\). Let \(r_0\) be a critical radius of \(\mathrm{Conf}_{k,r}(U)\), and take \(\gamma \neq0 \in \pi_m(\mathrm{Conf}_{k,r_0+\varepsilon}(U))\). What is the largest \(n\in\mathbb{N}\) for which there exists \(\gamma' \in \pi_m(\mathrm{Conf}_{n,r_0+\varepsilon})\) such that \((F_{n,k})_\ast(\gamma') = \gamma\) for all sufficiently small \(\varepsilon>0\)?
Having found some \(n\) and \(\gamma'\) satisfying the above conditions, we therefore have a non-zero \(\pi_m\)-class in \(\mathrm{Conf}_{n,r_0+\varepsilon}(U)\), where \(r_0\) is a higher critical radius for \(n\) discs. Furthermore, since \(F_{n,k} = F_{n,n'} \circ F_{n',k}\) for every \(k\le n' \le n\), this also yields a non-trivial \(\pi_m\)-class in each \(\mathrm{Conf}_{n',r_0+\varepsilon}(U)\), where \(r_0\) is a higher critical radius for every \(k< n' \le n\).
For the remainder of this paper, we consider the \(\pi_{k-3}\)-class in \(\mathrm{Conf}_{k, \frac{1}{k}+ \varepsilon}\) from §3. It is easy to find a lower bound on our question for these. We show in the proof of Thm. 15 that, as the angle \(\xi\) used in our construction tends to 0, the configuration is constrained ever more closely to \(H\mathrel{\vcenter{:}}= \bigcup_{i=1}^n B\left( (2i-n-1,0), \frac{1}{n} \right)\). Thus any disc \(D\) with \(\overline{D} \cap \overline{H} = \emptyset\) will not interfere with the given sphere of configurations for sufficiently small \(\xi\). Therefore, if we can fit \(l\) discs of radius \(r_0\) in \(D^2\backslash H_\varepsilon\) for some \(l\in \mathbb{N}\) and \(\varepsilon>0\), where \(H_\varepsilon\) denotes the \(\varepsilon\)-thickening of \(H\), then we have a lower bound \(n\ge k+l\).
We illustrate this with an example.
Proposition 17. Consider the non-trivial class \([\mathcal{S}] \in \pi_2(\mathrm{Conf}_{5, \frac{1}{5}+\varepsilon})\) constructed in §3. Then for \(5\le n\le19\), there is \([\mathcal{S}']\in \pi_2(\mathrm{Conf}_{n, \frac{1}{5}+\varepsilon})\) such that \((F_{n,5})_\ast([\mathcal{S}']) = [\mathcal{S}]\).
This is inspired by the packing of 19 discs in [18].
Proof. Consider the configuration \((D_1, \ldots, D_{19})\in \pi_2(\mathrm{Conf}_{19, \frac{1}{5}})\) in Fig. 6, in which \(x_i=(\frac{-6+2i}{5},0)\) for \(1\le i \le 5\) (five discs lined up across the horizontal diameter), and the remaining 14 discs are centred at the coordinates of form \(\frac{2}{5}\left(\cos\frac{\pi j}{3}, \sin\frac{\pi j}{3}\right)\) or \(\frac{4}{5}\left(\cos\frac{\pi j}{6}, \sin\frac{\pi j}{6}\right)\), \(j\) an integer.
When we roll \(D_{11}, D_{13}, D_{16}, D_{18}\) along \(\partial D^2\) towards the horizontal diameter until they touch their adjacent discs, they cease to touch \(D_6, D_7, D_8, D_9\). Thus we can move these four discs away from the horizontal diameter. Now all \(D_j\) lie outside a thickening of \(\bigcup_{i=1}^5 D_i\) for \(6\le j \le 19\). Therefore, if we choose the angles of \(\mathcal{S}\) to be sufficiently small, \(D_i\) (\(1\le i \le5\)) can trace out the positions of \(\mathcal{S}\) while the remaining discs are stationary. ◻
Since the best known packing of 20 discs into \(D^2\) has disc radius 0.19522 (see [3]), it appears likely that \(19\) is the best possible value of \(n\) for \(k=5\), as it is likely impossible to fit 20 discs of radius \(\frac{1}{5}+\varepsilon\) in \(D^2\) at all. On the other hand, it is not necessarily true that this technique gives the best possible lower bound for general \(n\), since this construction depends on a specific choice of representative of the homotopy class. However, in most representatives \(\mathcal{S}\) of this class, the variation of \(\bigcup \mathcal{D}\) over all \(\mathcal{D}\in \mathcal{S}\) is much greater. This means that the additional discs must be able to fit into many differently-shaped regions (given by \(D^2\backslash \bigcup \mathcal{D}\) for each \(\mathcal{D}\)), and must do so continuously, and without intersecting with each other. The configurations we constructed in §3 consist of an unbroken line of discs, in which the variation of \(\bigcup\mathcal{D}\) is symmetric about the \(x\)-axis. In this case, we see that for some \(\mathcal{D}\in \mathcal{S}\), there is more space above the configuration, and for other \(\mathcal{D}\in \mathcal{S}\), there is more space below, so some of the additional discs must move from one side of the configuration to the other. This may not be possible in a continuous way without intersecting other discs when we are near the packing limit. Thus it seems reasonable to conjecture the following:
Conjecture 18. Let \(0\neq \gamma \in \pi_{k-3}(\mathrm{Conf}_{k,\frac{1}{k}+\varepsilon})\) be the sphere found in §3. Let \(H\mathrel{\vcenter{:}}= \bigcup_{i=1}^k B\left( (2i-k-1,0), \frac{1}{k} \right)\), \(H_\varepsilon\) its \(\varepsilon\)-thickening, and \(m\) be the greatest number of discs of radius \(\frac{1}{k}\) which fit in \(D^2\backslash H_\varepsilon\) for some \(\varepsilon>0\). Then the answer to Question 16 is \(k+m\).
In complete generality, an upper bound is given by the maximal packing of discs of radius \(r_0+\varepsilon\) into \(D^2\) (\(\varepsilon>0\) arbitrarily small).