SU(2)/SL(2) knot invariants and KS monodromies

Abstract

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.