A polynomial action on colored ${\mathfrak{sl}}_2$ link homology

Abstract

We construct an action of a polynomial ring on the colored sl2 link homology of Cooper–Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar multiple of the ${\mathfrak{sl}}_2$ Reshetikhin–Turaev invariant. We expect this homology to be functorial under 4-dimensional cobordisms. The polynomial action is related to a conjecture of Gorsky–Oblomkov–Rasmussen–Shende on the stable Khovanov homology of torus knots, and as an application we obtain a weak version of this conjecture. A key new ingredient is the construction of a bounded chain complex which categorifies a scalar multiple of the Jones–Wenzl projector, in which the denominators have been cleared.