  In this paper we explore the braiding properties of the Moore-Read fractional
Hall sequence, which amounts to computing the adiabatic evolution of the Hall
liquid when the anyons are moved along various trajectories. In this work, the
anyons are pinned to precise spatial configurations by using specific external
potentials. Such external potentials break the translational symmetry and it
appears that one will be forced to simulate the braidings on the entire
many-body Hilbert space, an absolutely prohibitive scenario. We demonstrate how
to overcome this difficulty and obtain the exact braidings for fairly large
Hall systems. For this, we show that the incompressible state of a general
$(k,m)$ fractional Hall sequence can be viewed as the unique zero mode of a
specific Hamiltonian $H^{(k,m)}$, whose form is explicitly derived by using
k-particles creation operators. The compressible Hall states corresponding to
$n$$\times$$k$ anyons fixed at $w_1$,...,$w_{nk}$ are shown to be the zero
modes of a pinning Hamiltonian $H^{(k,m)}_{w_1,...,w_{nk}}$, which is also
explicitly derived. The zero modes of $H^{(k,m)}_{w_1,...,w_{nk}}$ are shown to
be contained in the space of the zero modes of $H^{(k,m)}$. Therefore, the
computation of the braidings can be done entirely within this space, which we
map out for a number of Hall systems. Using this efficient computational
method, we study various properties of the Moore-Read states. In particular, we
give direct confirmation of their topological and non-abelian properties that
were previously implied from the underlying Conformal Field Theory (CFT)
structure of the Moore-Read state.
