This response is for a general audience, so kindly do not be offended if I'm saying things you already know.

Here in plain old 3D toroidal vorticies are not unusual. There is the smoke ring. Scientists have made smoke ring cannon. Dolphins make toroidal vorticies of water with a core of air and use them for toys. I have also seen a video of such a toroidal vortex in the shape of a trefoil knot. It wasn't very stable but it did exist for a second or two.

There are also electromagnetic toroidal vorticies. Such are predicted to be rife in the superconductive cores of neutron stars. These vorticies have a twist which is topologically preserved so they should be fairly stable, though no one knows how much.

In some Type 1.5 superconductors it has been calculated that polygonal braided quantum toroidal vorticies will exist. That is, a "square" vortex has four subvorticies each with energy which is a fraction of a quantum. The four subvorticies are wrapped in a larger vortex which also has energy which is a fraction of a quantum. The sum of the energies of the five vorticies is an integer. It is then difficult for a subvortex to break down and release its energy, as the system cannot release a fraction of a quantum of energy. Such vorticies should be stable.

I learned all this from the work of Egor Babaev, who has had a great many articles published in Nature and Physical Review.

https://en.wikipedia.org/wiki/Egor_Babaev Among other things, he led experimentalists to prove that the Type 1.5 superconductor existed here on Earth.

Toroidal vorticies of light are theoretically possible. It has been claimed that they have been produced.

In 4D I would expect that toroidal vorticies would be more stable than in 3D. Going back to those 3D vorticies made of water and air, there is compression and expansion going on during the rotation. Indeed I find it surprising that such vorticies can persist for many seconds, but they do. In 4D there is no such compression and expansion, so it is a more favorable environment for such things.

In 4D it is normal to have two planes of rotation whose intersection is a point. The rotation in each plane is independent of the other. It seems that in this case one would need to think of two planar axies. In 4D a 2-plane does not partition the space, so you can't say whether a point outside the 2-plane is on one side or the other.