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

Abstract

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum ${\mathfrak{gl}}_N$ 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 $Y$.

arXiv comment43 pages

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Cited by in corpus (1)

• Inverted state sums, inverted Habiro series, and indefinite theta functions