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

Abstract: For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescu series denoted $F^{\mathfrak{g}}_K$ 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 $\mathfrak{g} = \mathfrak{sl}_2$ but comparatively less work has studied other lie algebras. In this paper we extend the large colour $R$ matrix from $\mathfrak{sl}_2$ to symmetrically coloured $\mathfrak{sl}_N$. This gives a definition for $F^{\mathfrak{sl}_N, sym}_K$ for positive 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 evidence towards a conjectural HOMFLY-PT analouge of $F_K$.