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

TL;DR

This paper extends the large colour $R$ matrix from $ ext{sl}_2$ to symmetrically coloured $ ext{sl}_N$, enabling computation of the Gukov-Manolescu series for a broader class of knots and links, and supporting a HOMFLY-PT conjecture.

Contribution

It introduces a new definition of the Gukov-Manolescu series for symmetrically coloured $ ext{sl}_N$, expanding computational and theoretical understanding beyond $ ext{sl}_2$.

Findings

Defines $F^{ ext{sl}_N, sym}_K$ for positive braid knots.

Predicts $F^{ ext{sl}_N, sym}_K$ for wider classes of knots and links.

Provides evidence for a HOMFLY-PT analogue of $F_K$.

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 $F_K$.