A categorification of quantum ${\mathfrak{sl}}_3$ projectors and the ${\mathfrak{sl}}_3$ Reshetikhin–Turaev invariant of tangles

Abstract

We construct a categorification of the quantum ${\mathfrak{sl}}_3$ projectors, the ${\mathfrak{sl}}_3$ analog of the Jones–Wenzl projectors, as the stable limit of the complexes assigned to $k$-twist torus braids (as $k\to \infty$) in a suitably shifted version of Morrison and Nieh’s geometric formulation of ${\mathfrak{sl}}_3$ link homology [14] We use these projectors to give a categorification of the ${\mathfrak{sl}}_3$ Reshetikhin–Turaev invariant of framed tangles.