Following the pioneering work of Kashaev many authors (Baseilhac-Benedetti, Bonahon-Liu, Bonahon-Wong-Yang, Dimofte-Garoufalidis, et al.) have constructed "cyclic-type" quantum hyperbolic invariants of knots and related objects. These are defined using discrete state-sums and are conjectured to exhibit asymptotic growth in the semiclassical limit. The whole zoo of invariants are conjectured (and in some cases known) to be closely related or equivalent. In recent work I defined yet another cyclic-type invariant (denoted Z_N^ψ) of links and tangles in S^3 using R-matrices defined in joint work with Reshetikhin. In this talk I will sketch the construction and explain the benefits of this approach. In particular, Z_N^ψ is defined without any phase ambiguity and bears strong formal resemblance to the geometric quantization of complex Chern-Simons theory. There is also a connection to the hyperbolic torsion at N = 2.