  We introduce the cutting construction of possibly non-compact symplectic
toric manifolds, in particular, toric symplectic cones that correspond to a
weakly convex good cone. Since the symplectization of a toric contact manifold
is a toric symplectic cone, we can also construct toric contact manifolds that
correspond to a weakly convex good cone by the cutting construction. (Note that
these toric contact manifolds can not be constructed by Delzant construction.)
We further prove there are no toric Sasakian structures on these contact
manifolds. From this, contact toric manifolds of toric K-contact type are of
toric Sasakian type.
