  We consider the issue of the slice invariance of refined topological string
amplitudes, which means that they are independent of the choice of the
preferred direction of the refined topological vertex. We work out two
examples. The first example is a geometric engineering of five-dimensional U(1)
gauge theory with a matter. The slice invariance follows from a highly
non-trivial combinatorial identity which equates two known ways of computing
the chi_y genus of the Hilbert scheme of points on C^2. The second example is
concerned with the proposal that the superpolynomials of the colored Hopf link
are obtained from a refinement of topological open string amplitudes. We
provide a closed formula for the superpolynomial, which confirms the slice
invariance when the Hopf link is colored with totally anti-symmetric
representations. However, we observe a breakdown of the slice invariance for
other representations.
