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 $F_K$.
Categories: math.GT hep-th math-ph math.MP math.QA