I'm afraid I'm not much more clued-in now about what you're thinking. I'll try my best.

> ... "braiding" of any system would contribute a (small?) mass to it -- and I wanted to see the calculation

My best guess is that you are thinking that the (far from active) black hole in the central parsec of our galaxy is somehow lifting mass out of itself and into the wider galaxy, and so will briefly discuss that.

A reasonable first step towards the theoretical footing behind that is Murata & Soda 2006, in Phys. Rev. D. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.74.04..., "Hawking radiation from rotating black holes and gravitational anomalies" (also at https://arxiv.org/abs/hep-th/0606069v2) where the scalar field (also seen in Hawking 1978) is literally a form of "anyon" field. I don't see how the use of the name "anyon" helps, however, and the outward flux is going to be small -- even very very small compared to the ordinary thermal collisions of gas and dust in the galaxy centre, let alone the stars there.

I think that means you're on course for an extension to the Standard Model of Particle Physics to add in some electromagnetically-non-interacting species that decays at some distance into electromagnetically-interacting ones, along the lines of various dark matter decay models, especially those designed to produce "feedback" in spite of quiet galactic centres. I don't think this is likely to bear fruit, but cf. this blog on DM->tau decay: http://honorsfellows.blogs.wm.edu/2011/06/12/decaying-dark-m...

> I assume I'm off on a wild chase, but I want to see where the math fails for my own education.

I'm afraid I can't join you on your chase, wild or not, but I think that the mathematics of black holes is reasonably accessible and easy enough to find in a variety of textbooks. Coupling an anyon field to it is an exercise in quantum field theory on curved spacetime (as in Murata & Soda) or perhaps a second-quantization of an electrovac solution based on Kerr-Newman. I'm really struggling to see how -- given the high temperatures and high particle numbers involved -- anything is to be gained by looking at the truly microscopic behaviour of the stress-energy tensor, even in the very near region of the horizon. I'm also struggling to see how such effects relating to our central black hole are not totally washed out by processes in the bulge or in the thin disc. Our central black hole is not only far from active, it is also quite small compared to the black holes we find in other galaxies, especially Seyferts and recently discovered high-redshift (z ~ 7.6) QSOs. There is also a lot of dust and gas along our line of sight to the galactic centre, and between these filamentary structures and the galactic centre. (A number of these filaments are much closer to known radio-bright supernova remnants, as detailed in the study, but don't seem very different from those far from known SNRs.) There is also no evidence for beyond-the-standard-model(-of-particle-physics) physics in the discovery of these filaments. I don't mind being asked to think about BTSM, but these filaments are a very poor justification for that.

Finally, the only knotting I expect around a quasar or microquasar are bright spots in the plasma of jets consistent with small-angle radiation, and it's hard to think of a better explanation than proper motion of the source. We see this in stellar-mass X-Ray binaries (especially nearby microquasars), for example. The absence in extragalactic quasars supports this idea, since the proper motion of those sources will necessarily be lower than galactic ones by a couple orders of magnitude.

> I want to see where the math fails for my own education

If you care to write down some math now, I promise to at least have a look at it and see if I can aim you at some additional resources which may be more helpful still.