I was wondering something similar few month ago. Then I concluded that most of the topological staffs appear at the boundary between two different topological sector. A sector being characterised by a Chern number, or if you prefer a topological charge, one needs a boundary / an interface between two systems characterised by different topological charge.
A $k$-space (or momentum, or reciprocal, or Fourier, ...) is well defined only for periodic boundary conditions. The fact that the $x \leftrightarrow k$ is a Fourier transform imposes a periodicity in $x$ or in $k$. That's the stringent condition under which $k$ is a good quantum number. Note that we can still define some quasi-$k$ for disordered media. So we could not in principle define a $k$-space when a system has boundary. Note that infinite system are usually closed by periodic boundary condition, also called Born-von-Karman conditions.
I'm not aware so much about anyons (I'm still learning about that) but I believe they (almost all of them ? all of them ? I don't know) appear due to boundary conditions in condensed matter, for the reason I gave about the topological charge transition. So I believe it should be impossible to define anyons in $k$-space, for the simple reason that the $k$-space is not a correct description of the matter when anyons exist.
I would really appreciate comments/critics about what I said, especially if it's (partially) wrong.