The Alexander polynomial as quantum invariant of links

TL;DR

This paper explores the quantum invariants derived from the representation theory of U_q(gl(1|1)), demonstrating how to recover the Alexander polynomial as a specific case, and clarifying the structure of related invariants.

Contribution

It provides an explicit description of the ribbon structure on U_q(gl(1|1)) representations and shows how to obtain the Alexander polynomial from this quantum invariant.

Findings

Explicit ribbon structure on U_q(gl(1|1)) representations

Construction of quantum invariants of framed tangles

Derivation of the Alexander polynomial from the vector representation

Abstract

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 $U_q(\mathfrak{gl}(1|1))$.