We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined via planar state sums based on link diagrams and a new family of enhanced Yang-Baxter operators (YBO) that we compute explicitly. By a local comparison of the respective YBO's we show that these invariants coincide with Kashaev's specializations of the colored Jones polynomials. As a further application we disprove a conjecture about the semi-classical limits of quantum hyperbolic invariants, by showing that it conflicts with the existence of hyperbolic links that verify the volume conjecture.