Flavor invariants for the SM with one singlet vector-like quark

We study the flavor invariants of the SM augmented by one singlet vector-like quark. Aided by the Hilbert series, we construct all the basic invariants with which any flavor invariant can be written as a polynomial. In special, this theory contains one CP odd invariant of degree six which has degree much lower than the usual Jarlskog invariant of the SM. We find the nonlinear polynomial relations (syzygies) of lowest degrees involving these basic invariants, including the expression of the square of the CP odd invariant of lowest degree in terms of CP even invariants. The $SU(3)$ identity underlying this syzygy is uncovered in terms of invariant tensors, which can be applied to rewrite any square of a CP odd invariant of the same form, involving three hermitean matrices of size three. We demonstrate by an example that there is CP violation that is not detected by the CP odd invariants proposed in the literature so far but it can be detected with the full list of CP odd invariants found here.