Mathematicians have found a hidden 'reset button' for undoing rotation

There's a bit of a caveat with the anti-twister mechanism, namely, that the wiring must be loose enough to pass around the supplied rotating part.

This is important. The mechanism doesn't really work the way you want most of the time. I occasionally see a claim that you can power a carousel with this method, but it doesn't work. You would have to have the cable go out and around the carousel structure, and then into the top. And the cable would still move relative to the ground and the carousel.

You could, in principle, have a totally internal system, but with arms that grab and release the cable at intervals so that the looped portion can pass by them. You could arrange the timing so that electrical contact is never lost. But you are still making/breaking contact and it starts to lose some apparent advantages compared to a slip ring.

That's not to say it isn't still useful for some purposes, like maybe a radio antenna that isn't too impacted by a cable moving in front on occasion. But it doesn't eliminate all uses for a slip ring.

And no axle to rotate on.

Wouldn't a slip ring help here?

I can't go into detail, but that's essentially my use case. I have a geodesic dome with a cable running up externally, and would like to run it through a hollow shaft coming in through the top which rotates like a carousel. I'm fairly certain this is precisely what I need.

The whole point of the anti-twister mechanism is that it doesn't use a slip ring.