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, 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.

31 pages. Published version with an additional appendix

References in corpus (8)

• The Refined Topological Vertex
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Link Homologies and the Refined Topological Vertex
• Matrix Factorizations and Kauffman Homology
• Changing the preferred direction of the refined topological vertex
• Flop Invariance of Refined Topological Vertex and Link Homologies
• Non-Symmetric Hall-Littlewood Polynomials
• A Commutative Family of Integral Transformations and Basic Hypergeometric Series. II. Eigenfunctions and Quasi-Eigenfunctions

Cited by in corpus (19)

• Proving AGT conjecture as HS duality: extension to five dimensions
• Fiber-Base Duality and Global Symmetry Enhancement
• 5d/6d DE instantons from trivalent gluing of web diagrams
• On 3d extensions of AGT relation
• Changing the preferred direction of the refined topological vertex
• Sekiguchi-Debiard operators at infinity
• Hopf link invariants and integrable hierarchies
• The MacMahon R-matrix
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Kerov functions for composite representations and Macdonald ideal
• Refined Topological Strings on Local ${\mathbb{P}}^2$
• Challenges of beta-deformation
• A non-torus link from topological vertex
• Can tangle calculus be applicable to hyperpolynomials?
• On the status of DELL systems
• Lax operator for Macdonald symmetric functions
• K-Theory of Hilbert Schemes as a Formal Quantum Field Theory
• Hopf superpolynomial from topological vertices