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

Abstract

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