Cyclotomic expansions for glN knot invariants via interpolation Macdonald polynomials

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum glN in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the glN Reshetikhin--Turaev link invariants and the universal glN knot invariant; 2) an explicit construction of the unified glN invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for sl2. In addition, we give a simple proof of the fact that the universal glN invariant of any evenly framed link and the universal slN invariant of any 0-framed algebraically split link are Γ-invariant, where Γ=Y/2Y with the root lattice Y