It is well known that the Alexander polynomial of a link can be constructed as the quantum invariant associated to the quantum enveloping superalgebra \(U_q(\mathfrak{gl}(1|1))\) [L.H. Kauffman andH. Saleur, Commun. Math. Phys. 141, No. 2, 293–327 (1991;Zbl 0751.57004)].
Author’s summary: “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 \(U_q(\mathfrak{gl}(1|1))\)-representations 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 non-zero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of \(U_q(\mathfrak{gl}(1|1))\).”

Reviewer: Kazuo Habiro (Kyoto)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.