Cyclotomic expansions for \(\mathfrak {gl}_N\) link invariants via interpolation Macdonald polynomials

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