# Projection of Antoine's necklace

Antoine's necklace is a pathological embedding of the Cantor set into $\Bbb R^3$. The second iteration looks like this:

Interestingly, the complement $\Bbb R^3\setminus\rm A$ is not simply connected. This property is preserved by ambient isotopies. (Thanks, @MikeMiller!) Anything with an ambient isotopy to what's defined in the article, then, should be considered to be an Antoine's necklace.

What happens when you project the necklace onto a plane? That is, if $\pi:\Bbb R^3\to\Bbb R^2$ is a projection, what is $\pi({\rm A})$? I'm sure you don't get another Cantor set. In fact, it seems like it would always be connected. My guess is that it must be homeomorphic to the Sierpiński carpet. Is it?

• An ambient isotopy induces a homeomorphism of the complement of $A$, so yes, that's correct.
– user98602
Mar 2 '16 at 15:46
• I think that a projection of an Antoine necklace onto the plane might be homeomorphic to the Sierpinski carpet, but I don't believe that it must be. Keep in mind that Antoine's necklace is a topological entity - unless you specify a specific configuration using, for example, an iterated function system. Thus, there's quite a bit of flexibility as to how it is laid out in space. Furthermore, it is certainly possible for the projection of a Cantor set in $\mathbb R^3$ to contain a solid disk. I imagine that a projection of an Antoine necklace might contain a solid disk as well. Mar 4 '16 at 23:30
• @MarkMcClure I'll define "an Antione's necklace" to be anything with an ambient isotopy to the one defined in the article (which I'll make more explicit in the question) I'm not sure how a projection of such a thing would contain a disk, though I suppose it might be possible. Mar 5 '16 at 23:46
• On a computer screen, isn't any drawing of a 3D object just a projection of that object onto a plane? So if you want intuition about your projection, drawings of Antoine's necklace from different angles with higher iterations may be helpful. This is a video with enough iterations to reach 160,000 tori. Mar 6 '16 at 0:03
• @AkivaWeinberger For every $n$, there is a Cantor set in $\mathbb R^n$ whose projection onto every $n-1$ dimensional subspace contains an $n-1$ dimensional ball. See, for example, This sets with fat shadows in the Monthly. Now, I'm not assuring you that an Antoine's necklace can be so configured but, if you're going to allow it to be twisted any which way, I don't see any reason to think that it can't. Mar 6 '16 at 0:38