  In this sequel to earlier papers by three of the authors, we obtain a new
bound on the complexity of a closed 3--manifold, as well as a characterisation
of manifolds realising our complexity bounds. As an application, we obtain the
first infinite families of minimal triangulations of Seifert fibred spaces
modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$
