The ${\mathfrak{s}\mathfrak{l}}_N$ Symmetrically Large Coloured $R$ Matrix

For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescu series denoted $F_K^{\mathfrak{g}}$ which serves as an analytic continuation of the quantum knot invariants associated to finite dimensional irreducible representations of $\mathfrak{g}$. There has been a great deal of work done on computing this invariant for $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ but comparatively less work has studied other lie algebras. In this paper we extend the large colour $R$ matrix from ${\mathfrak{s}\mathfrak{l}}_2$ to symmetrically coloured ${\mathfrak{s}\mathfrak{l}}_N$. This gives a definition for $F_K^{{\mathfrak{s}\mathfrak{l}}_N,sym}$ for positive braid knots and allows for predictions of $F_K^{{\mathfrak{s}\mathfrak{l}}_N,sym}$ for a much larger class of knots and links. It also provides further evidence towards a conjectural HOMFLY-PT analouge of $F_K$.