Categorified $\mathfrak{sl}_N$ invariants of colored rational tangles

Abstract

We use categorical skew Howe duality to find recursion rules that compute categorified ${\mathfrak{s}\mathfrak{l}}_N$ invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links and discuss consequences for the corresponding decategorified invariants.