Cyclotomic expansions for $\mathfrak{gl}_N$ knot invariants via interpolation Macdonald polynomials

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ 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 $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for $\mathfrak{sl}_2$. In addition, we give a simple proof of the fact that the universal $\mathfrak{gl}_N$ invariant of any evenly framed link and the universal $\mathfrak{sl}_N$ invariant of any $0$-framed algebraically split link are $\Gamma$-invariant, where $\Gamma=Y/2Y$ with the root lattice $Y$.