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For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescuseries denoted $F^{\mathfrak{g}}_K$ which serves as an analytic continuation ofthe quantum knot invariants associated to finite dimensional irreduciblerepresentations of $\mathfrak{g}$. There has been a great deal of work done oncomputing this invariant for $\mathfrak{g} = \mathfrak{sl}_2$ but comparativelyless work has studied other lie algebras. In this paper we extend the largecolour $R$ matrix from $\mathfrak{sl}_2$ to symmetrically coloured$\mathfrak{sl}_N$. This gives a definition for $F^{\mathfrak{sl}_N, sym}_K$ forpositive braid knots and allows for predictions of $F^{\mathfrak{sl}_N, sym}_K$for a much larger class of knots and links. It also provides further evidencetowards a conjectural HOMFLY-PT analouge of $F_K$.

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