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

### Anna Beliakova, Eugene Gorsky

**20/1/21**Published in : arXiv:2101.08243

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). Two main applications are: 1) a cyclotomic expansion of the universal \mathfrak{gl}_N knot invariant and 2) an explicit construction of the unified \mathfrak{gl}_N invariants for integer homology 3-spheres obtained by knot surgeries. 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.