It’s actually drawn thickened up a bit and made transparent since I really want to look at the space around the link, the Whitehead link exterior. Also shown is an axis for a 180 degree spin, an involution.

Quotient out by this involution and you get a tangle in $S^2 \times I$.

The Whitehead link exterior becomes the $S^2 \times I$ and the involution axis becomes the strands of the tangle.

Taking the double branched cover gets you back to the space outside the Whitehead link.

Plugging up the outer sphere with a particular rational tangle makes the whole thing a rational tangle.

We’re inside the ball of the rational tangle and looking out to that small sphere:

Shrink the “ears” and slip off the spiral to see it unravel…

But this isn’t what’s so interesting.

There’s this thing called the Whitehead manifold. While asking Google to tell me the Wikipedia link again I noticed this post that appears to give a decent overview of the Whitehead manifold.

Basically -and omitting a few details- the Whitehead manifold is obtained by stacking together an infinite ray of Whitehead link exteriors, but starting the whole thing off with a solid torus. Let’s see the corresponding thing with tangles. (Note that I neglected to plug the beginning end up with the rational tangle.)

Here’s two stacked together.

Here’s a whole bunch more together.

You’ll notice the 90 degree rotation as they get stacked. That’s so if I were to plug up the beginning as before, stopping the stack at any finite depth would yield a rational tangle. Doing the rotating to the right or left at each step lets you make lots of different tangles of a given finite depth whose double branched covers will be the same stack of Whitehead link exteriors. Since we’re doing an infinite stack, this makes the Whitehead manifold double branch cover uncountably many different such tangles.

Okay, so maybe some of those statements should be checked… by someone else. And these are wild tangles rather than the usual sort of tangles. But that’s okay. Here are some more pics.