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In this paper we construct a new basis for the cyclotomic completion of thecenter of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonaldpolynomials. Then we use a result of Okounkov to provide a dual basis withrespect to the quantum Killing form (or Hopf pairing). The main applicationsare: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaevlink invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) anexplicit construction of the unified $\mathfrak{gl}_N$ invariants for integralhomology 3-spheres using universal Kirby colors. These results generalize thoseof Habiro for $\mathfrak{sl}_2$. In addition, we give a simple proof of thefact that the universal $\mathfrak{gl}_N$ invariant of any evenly framed linkand the universal $\mathfrak{sl}_N$ invariant of any $0$-framed algebraicallysplit link are $\Gamma$-invariant, where $\Gamma=Y/2Y$ with the root lattice$Y$.

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