Macdonald operators and homological invariants of the colored Hopf link

Abstract

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozçaz 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.