Abstract
We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of $U_q(\mathfrak{gl}(1|1))$. This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family of relations. The technique also gives a useful connection between a system of symmetries on $\bigoplus_m \dot{U}_q(\mathfrak{gl}(m))$ and the braiding on the category of $U_q(\mathfrak{gl}(1|1))$-representations which can be used to construct the Alexander polynomial and coloured variants.
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