Taming Triangulation Dependence of $T$$^{6}$/$\mathbb{Z}$$_{2}$ x $\mathbb{Z}$$_{2}$ Resolutions

Resolutions of certain toroidal orbifolds, like $T$$^{6}$/$\mathbb{Z}$$_{2}$ x $\mathbb{Z}$$_{2}$, are far from unique, due to triangulation dependence of their resolved local singularities. This leads to an explosion of the number of topologically distinct smooth geometries associated to a single orbifold. By introducing a parameterisation to keep track of the triangulations used at all resolved singularities simultaneously, (self-)intersection numbers and integrated Chern classes can be determined for any triangulation configuration. Using this method the consistency conditions of line bundle models and the resulting chiral spectra can be worked out for any choice of triangulation. Moreover, by superimposing the Bianchi identities for all triangulation options much simpler though stronger conditions are uncovered. When these are satisfied, flop--transitions between all different triangulations are admissible. Various methods are exemplified by a number of concrete models on resolutions of the $T$$^{6}$/$\mathbb{Z}$$_{2}$ x $\mathbb{Z}$$_{2}$ orbifold.