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

We construct the invariant $F_K^{{\mathfrak{s}\mathfrak{l}}_3}\in \mathbb{Z}[q,q^{-1}][[x,y]]$ 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 ${\mathfrak{s}\mathfrak{l}}_3$ to all irreducible representations. We conclude with a conjectural framework for constructing $F_K^{{\mathfrak{s}\mathfrak{l}}_N}$ for arbitrary $N$.