A Non-Abelian Generalization of the Alexander Polynomial from Quantum sl₃

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum sl₂ at a fourth root of unity. We generalize this construction to define a link invariant Δg for any semisimple Lie algebra g of rank n, taking values in n-variable Laurent polynomials. Focusing on the case g=sl₃, we establish a direct relation between Δsl₃ and the Alexander polynomial. We show that certain parameter evaluations of Δsl₃ recover the Alexander polynomial on knots, despite the R-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate Δsl₃ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.