  Murakami and Ohtsuki have shown that the Alexander polynomial is an
$R$-matrix invariant associated with representations $V(t)$ of unrolled
restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. In this
context, the highest weight $t\in\mathbb{C}^\times$ of the representation
determines the polynomial variable. For any semisimple Lie algebra
$\mathfrak{g}$ of rank $n$, we extend their construction to a link invariant
$\Delta_{\mathfrak{g}}$, which takes values in $n$-variable Laurent
polynomials. The focus of this paper is the case
$\mathfrak{g}=\mathfrak{sl}_3$. For any knot $K$, evaluating
$\Delta_{\mathfrak{sl}_3}$ at ${t_1=1}$, ${t_2=1}$, or ${t_2= it_1^{-1}}$
recovers the Alexander polynomial of $K$. This is not obvious from an
examination of the $R$-matrix, as the $R$-matrix evaluated at these parameters
does not satisfy the Alexander-Conway skein relation. We tabulate
$\Delta_{\mathfrak{sl}_3}$ for all knots up to seven crossings along with
various other examples. In particular, it distinguishes the Kinoshita-Terasaka
knot and Conway knot mutant pair and is nontrivial on the Whitehead double of
the trefoil.

Quantum Algebra;Geometric Topology;Representation Theory