The unlinking torus

A little known fact: If you have a torus (a donut-shaped tube) with a hole in it, then it is possible to turn it inside-out. I will leave the details for you to work out, but rest assured that it is possible.

That's the inspiration for this puzzle (which I basically stole from Martin Gardner).

Here we have a torus. There's a big hole in it so that we can turn it inside-out. There is also a ring (in red) painted on the outside of the torus. There is also a ring (in dark blue) painted on the inside of the torus. These two rings are linked.

But suppose I turned the tube inside-out. Then the red ring will be on the inside, while the blue ring will be on the outside. Then the rings will no longer be linked!

Did I really manage to unlink the rings? If so, how?

See the solution