  We consider the k=1 Friedman-Robertson-Walker (FRW) model within loop quantum
cosmology, paying special attention to the existence of an ambiguity in the
quantization process. In spatially non-flat anisotropic models such as Bianchi
II and IX, the standard method of defining the curvature through closed
holonomies is not admissible. Instead, one has to implement the quantum
constraints by approximating the connection via open holonomies. In the case of
flat k=0 FRW and Bianchi I models, these two quantization methods coincide, but
in the case of the closed $k$=1 FRW model they might yield different quantum
theories. In this manuscript we explore these two quantizations and the
different effective descriptions they provide of the bouncing cyclic universe.
In particular, as we show in detail, the most dramatic difference is that in
the theory defined by the new quantization method, there is not one, but two
different bounces through which the cyclic universe alternates. We show that
for a `large' universe, these two bounces are very similar and, therefore,
practically indistinguishable, approaching the dynamics of the holonomy based
quantum theory.
