  In these notes we collect some results about finite dimensional
representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed
tangles which are well-known to experts but difficult to find in the
literature. In particular, we give an explicit description of the ribbon
structure on the category of finite dimensional
$U_q(\mathfrak{gl}(1|1))$-representation and we use it to construct the
corresponding quantum invariant of framed tangles. We explain in detail why
this invariant vanishes on closed links and how one can modify the construction
to get a nonzero invariant of framed closed links. Finally we show how to
obtain the Alexander polynomial by considering the vector representation of
$U_q(\mathfrak{gl}(1|1))$.
