Macdonald operators and homological invariants of the colored Hopf link

TL;DR

This paper explores the homological invariants of the colored Hopf link using Macdonald operators, proving polynomiality and deriving explicit formulas, while suggesting modifications for non-negative integer coefficients.

Contribution

It introduces a power sum realization for Macdonald operators to analyze the GIKV proposal, proving polynomiality and providing explicit formulas for antisymmetric representations.

Findings

Proved polynomiality of GIKV invariants for all representations

Derived a closed formula for antisymmetric representations

Suggested amendments to GIKV's proposal for non-negative coefficients

Abstract

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV's proposal is required to make all the coefficients of the polynomial non-negative integers.