Manifestly gauge-independent formulations of the Z2 invariants

We use a monodromy argument to derive new expressions for the Z(2) invariants of topological insulators with time-reversal symmetry in two and three dimensions. The derivations and the final expressions do not require any gauge choice and the calculation of the invariants is based entirely on the projectors onto the occupied states. Explicit numerical tests for tight-binding models with strongly broken inversion symmetry are presented in two and three dimensions.