  A model describing the three-dimensional folding of the triangular lattice on
the face-centered cubic lattice is generalized allowing the presence of defects
corresponding to cuts in the two-dimensional network. The model can be
expressed in terms of Ising-like variables with nearest-neighbor and plaquette
interactions in the hexagonal lattice; its phase diagram is determined by the
Cluster Variation Method. The results found by varying the curvature and defect
energy show that the introduction of defects turns the first-order crumpling
transitions of the model without defects into continuous transitions. New
phases also appear by decreasing the energy cost of defects and the behavior of
their densities has been analyzed.
