Antoine's Necklace

Visualization

@rotx
$a=h
a=0
ret=[1,0,0, 0,cos(a),-sin(a), 0,sin(a),cos(a)]

@roty
$a=h
a=0
ret=[cos(a),0,-sin(a), 0,1,0, sin(a),0,cos(a)]

@
$n=Antoine's necklace
$dim=3
pi=2*asin(1)
r=rotx(pi/6)
s1=[0,1,0]*0.2
s2=rotx(pi/12)*[0,1,0]*0.2*rotx(pi/2)*roty(pi/2)
S=(1|r|r^2|r^3|r^4|r^5|r^6|r^7|r^8|r^9|r^10|r^11)*(s1|s2)
K=S*K

Overview

Antoine’s necklace is a subset of $\mathbb{R}^3$ that is homeomorphic to the standard Cantor set, yet whose complement $\mathbb{R}^3 \setminus K$ is not simply connected. It was constructed by Louis Antoine in 1921 as an example showing that the Schoenflies theorem fails in three dimensions.

The construction begins with a solid torus $T \subset \mathbb{R}^3$. Inside $T$ one places a simple chain — a finite collection of smaller solid tori $T_1, \ldots, T_k$ ($k > 3$) whose centres lie on the core circle of $T$, such that $T_i$ and $T_j$ are linked if and only if $|i - j| \equiv 1 \pmod{k}$, and otherwise disjoint. The process is then repeated inside each $T_i$, and the necklace $K$ is the intersection over all stages:

$K$ is a compact, totally disconnected, perfect subset of $\mathbb{R}^3$ — a Cantor set — but any loop linking $T$ cannot be contracted to a point in $\mathbb{R}^3 \setminus K$, unlike the complementary situation for the standard Cantor set in $\mathbb{R}^3$.

Self-Similar Construction

A necklace is called self-similar if there exists an IFS $\{S_1, \ldots, S_k\}$ of contracting similarities such that $S_1(T), \ldots, S_k(T)$ form a simple chain inside $T$ and the attractor satisfies $K = \bigcup_{i=1}^k S_i(K)$.

It is called regular if additionally $k$ is even and all maps share the same contraction ratio. Željko (2005) proved that a regular self-similar Antoine necklace requires at least $k = 22$ tori; the smallest known even value satisfying the geometric constraints is $k = 24$.

This Example

This entry uses $k = 24$ links arranged in two interleaved rings of 12. The 24 maps are grouped as:

where $r$ is a rotation by $\pi/6$ and $s_1$, $s_2$ place two tori at each of the 12 rotational positions. Each map contracts by a factor of $0.2$, giving an estimated Hausdorff dimension of:

Different values of $k$ (Sher 1968) yield Antoine necklaces with non-equivalent topological embeddings in $\mathbb{R}^3$ — inequivalence here means there is no self-homeomorphism of $\mathbb{R}^3$ carrying one onto the other.

References

• Antoine, L. (1921). Sur l'homeomorphisme de deux figures et leurs voisinages. J. Math. Pures Appl., 4, 221–325.
• Sher, R. B. (1968). Concerning Wild Cantor Sets in E³. Proc. Amer. Math. Soc., 19(5), 1195–1200.
• Željko, M. (2005). Minimal number of tori in geometric self-similar Antoine Cantor sets. JP J. Geom. Topol., 5(2), 239–245.
• Antoine's necklace — Wikipedia

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