I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander polynomial of a knot in terms of the representation theory of $\mathcal{U}_q(\mathfrak{gl}_{1|1})$ and (alternatively) $\mathcal{U}_q(\mathfrak{sl}_2)$ with $q$ a fourth root of unity.

But the state-sum model I know and love for the Alexander polynomial is Kauffman's original one (from his book "Formal Knot Theory"). Is this also derived from representation theory, like in Viro's paper (for one of these algebras or some other one)?

I'm interested in this question because the Kauffman state model is useful in knot Floer homology, which categorifies the Alexander polynomial. As far as I know (e.g. Sartori's recent preprint arXiv:1305.6162), a representation-theoretic categorification of the Alexander polynomial, thought of as a quantum $\mathfrak{gl}_{1|1}$-invariant, doesn't yet exist but may exist soon. I'd be very curious to know what sort of representation-theoretic structures knot Floer homology corresponds to, but as this seems to be a more difficult question, I'll stick with the Kauffman states for now!