  We review the Reshetikhin-Turaev approach to construction of non-compact knot
invariants involving R-matrices associated with infinite-dimensional
representations, primarily those made from Faddeev's quantum dilogarithm. The
corresponding formulas can be obtained from modular transformations of
conformal blocks as their Kontsevich-Soibelman monodromies and are presented in
the form of transcendental integrals, where the main issue is manipulation with
integration contours. We discuss possibilities to extract more explicit and
handy expressions which can be compared with the ordinary (compact) knot
polynomials coming from finite-dimensional representations of simple Lie
algebras, with their limits and properties. In particular, the quantum
A-polynomials, difference equations for colored Jones polynomials should be the
same, just in non-compact case the equations are homogeneous, while they have a
non-trivial right-hand side for ordinary Jones polynomials.
