# packing numbers and configuration spaces of the torus

Let $$S^1$$ be the unit circle of radius $$1$$.

For any $$k\geq 1$$, let the $$k$$-dimensional torus $$T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$$ be the $$k$$-fold self-Cartesian product of $$S^1$$.

Equip $$T^k$$ with the product Riemannian metric.

For any real number $$r>0$$ and any integer $$n\geq 1$$, consider the (ordered) disk configuration space

$$F_r(T^k,n)=\Big\{(x_1,x_2,\ldots,x_n)\in T^k~~\Big |~~ d(x_i,x_j)\geq 2r ~{\rm ~for~any~}~1\leq i

Fix $$r$$ and $$k$$.

If $$n$$ is large enough, then $$F_r(T^k,n)=\emptyset$$.

Denote the largest $$n$$ such that $$F_r(T^k,n)\neq \emptyset$$ by $$N(k,r)$$.

Question 1. For any integer $$k\geq 1$$ and any real number $$r>0$$, how to compute $$N(k,r)$$?

Question 2. For any real number $$r>0$$, how to compute the limit $$\lim_{k\to\infty }N(k,r)^{\frac{1}{k}}$$?

Are there any references?

Thank you very much.

• How is the problem related to braid groups? Commented Dec 15, 2021 at 6:28
• Hi Prof. I am not sure how they are related. @Mr. Commented Dec 15, 2021 at 7:37
• How is this related to sphere packing in Euclidean space? For example, I imagine this can be answered for certain $k$ and $r$ by looking at periodic closely-packings of spheres in Euclidean space. Commented Dec 15, 2021 at 8:43

If you rescale your circles to have circumference 2, then you are interested in packings of $$N$$ spheres in $$n$$-dimensional Euclidean space with periodicity $$(2\mathbb{Z})^n$$. The configuration space is usually written as $$\mathbb{R}^n/(2\mathbb{Z})^n$$, and the minimum distance between spheres, $$2r$$, would have the restriction $$r\le 1$$ so that the coset elements also respect the packing constraint. The parameter $$r$$ is the sphere radius.
For small $$n$$ and $$r=1$$ the maximum $$N$$ is achieved by Hamming distance 4 binary codes, where all the coordinates are 0 and 1. You can find the best known $$N$$ in the first column of the table on this website: