A large color $R$-matrix for $\mathfrak{sl}_3$

Abstract: We construct the invariant $$F_K<sup>{\mathfrak{sl}_3}\in\mathbb{Z}[q,q<sup>{-1}][[x,y]]$$ for any positive braid knot $K$, whose existence was conjectured by Park, building on earlier work of Gukov--Manolescu. The main step in our work extends a result by the first author on the invariant associated to symmetric representations of $\mathfrak{sl}_3$ to all irreducible representations. We conclude with a conjectural framework for constructing $F_K<sup>{\mathfrak{sl}_N}$ for arbitrary $N$.