Quantum relatives of the Alexander polynomial

The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin–Turaev functors based on irreducible representations of quantized $\mathrm{gl}(1|1)$ and $\mathrm{sl}(2)$. The corresponding face state sum models for the generalized Conway function are presented.