Topological invariants from non-restricted quantum groups

gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in [12] and [17], respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over non-restricted quantum groups considere