# Experimental Hopf fibrations

Recently I read a paper where the authors experimentally constructed a Hopf fibration - that is, they created a quantum system where the nematic vector field of the system had a non-zero Hopf invariant, and therefore corresponded to a non-trivial element of $\pi_3 (S^2)$.

I've read on wikipedia that the Hopf fibration can also be exhibited as a solution to the (compressible, non-viscous) Navier-Stokes equations. That is, there is a solution in which the fluid flows along the circles of the projection of the Hopf fibration to three dimensional space.

Question: Is there an experimental realisation of this fluid dynamics solution, or a reason why we can't construct one?

More generally, are there other examples of Hopf fibrations occurring in physics? I'm mostly interested in experimental or observational examples rather than theoretical constructions. So, for example, although a magnetic monopole is a wonderful example of the Hopf fibration occurring in physics, it's not what I'm looking for here.

• Do Hopf bifurcations count? If so, some particle orbits/trajectories in the terrestrial radiation belts can be modeled as Hopf bifurcation orbits. – honeste_vivere Feb 1 '16 at 19:03
• Nope. The two aren't related (they aren't even from the same Hopf, as I just found out from wikipedia). – Mark B Feb 1 '16 at 19:08
• Ah I see... Interesting that two different Hopf's came up with similar sounding theoretical concepts ;) – honeste_vivere Feb 1 '16 at 19:37
• Technically there are no exact examples of Hopf fibrations or anything else that mathematics has to offer in physics. Physicists are happy when they see something in approximation. While this doesn't address your question, it's important to keep that in mind when you are looking for the implementation of a mathematical object in the real world. It's all just $\pi \times thumb$. – CuriousOne Feb 1 '16 at 19:55