Colored Torus Link Homology

The package of Type A link homology theories is controlled by (Type A) singular Soergel bimodules. Weintroduce several families of “Fray” functors on singular Soergel bimodules, modeled on constructions incategorical representation theory. We prove that each of these families gives rise to a corresponding familyof column-colored link homologies interpolating between existing constructions of column-colored homology.Finally, we use Fray functors to compute the column-colored HOMFLY homology of positive torus knots,resulting in the first verification of mirror symmetry conjectures for colored HOMFLY homology in nontrivialcolors.