The Alexander polynomial/Conway potential can be computed as the quantum invariant associated to (a certain quotient of) $\mathcal U_q(\mathfrak{sl}_2)$ for $q = i$. More generally this works for $q = \xi = e^{\pi i / r}$ a primitive $2r$th root of unity. The $r = 2$ case gives the Alexander polynomials, while $r > 2$ are known as ADO invariants [1] or colored Alexander invariants (with $r$ the color.) Another reference is [2].

EDIT: Based on Noah's comments, I'll be a little more specific about the construction. At $q = \xi$ a $2r$th root of unity, the elements $K^r, E^r, F^r$ become central. For the ADO invariants we want representations of $$\mathcal A = \mathcal{U}_\xi(\mathfrak{sl}_2)/(K^r - t, E^r, F^r)$$ where $t$ is an indeterminate (which will become the variable of the Alexander polynomial.) The $r$th ADO invariant of a link is the Reshetikhin-Turaev invariant assigning each strand to an $r$-dimensional representation of $\mathcal A$. (Actually, the quantum dimensions vanish, so you need a slight generalization of the RT construction. Also, there are multiple non-isomorphic $r$-dimensional irreps, but they are very closely related.)

There are few references computing examples of these invariants. In particular, I'm curious to know if anyone's computed a skein relation for them.

- Y. Akustu, T. Deguchi, T. Ohtsuki,
*Invariants of colored links*, J. Knot Theory Ramifications1 (1992) 161184. - Cristina Ana-Maria Anghel,
*A topological model for the coloured Alexander invariants*arXiv:1906.04056