  Cyclotron braid subgroups are defined in order to identify the topological
origin of Laughlin correlations in 2D Hall systems. Flux-tubes and vortices for
composite fermion constructions are explained in terms of unavoidably
multilooped cyclotron braids. A link of braid picture with quasiclassical
quantum dynamics is conjectured in order to support the phenomenological model
of composite fermions with auxiliary fluxtubes, for Landau level fillings out
of 1/p, p odd. The even denominator fractional lowest Landau level fillings,
including Hall metal at n = 1/2, are also discussed in cyclotron braid terms.
The topological arguments are utilized to explain novel experimentally observed
features of the fractional quantum Hall state in graphene including the
triggering role of carriers mobility for this collective state.
