The Alexander polynomial as quantum invariant of links

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 $U_q(\mathfrak{gl}(1|1))$.Comment: This is a corrected and revised version, to be published on Arkiv f\"or Matematik. This is part of the author's PhD Thesi

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

“…The resolution of this problem is to choose a basepoint on the link, and cut the link at that point to yield a .1; 1/-tangle. The idea of cutting basepoints to obtain link invariants is well-established, and appears for example in [19] and explored in a more general context in [7]. Then we can apply the MOY moves until we have a polynomial times a single strand, and define .L/ to be this polynomial.…”

“…The ribbon element v acts as the identity on C 1j1 q and the element u (with S 2 .x/ D uxu 1 for any x ) acts as K (see, for example, [19] or [23] for more details).…”

“…In fact, one finds that .L/ is a Laurent polynomial and is independent of the choice of tensor factors involved in the trace, and the braid presentation of L (this is essentially [19,Theorem 4.6]). …”

“…This is only possible by investigating either a quantum universal enveloping algebra with the value q taken to be a root of unity, or by slightly extending the work of Reshetikin and Turaev to superalgebras, where the extra Z=2Z grading provides a setting where the quantum dimension can be 0. A thorough description of this is given in Sartori [19].…”

Determinant Formulae of Matrices with Certain Symmetry and Its Applications

“…The Alexander polynomial is categorified by Knot Floer homology, see [14,15]. Furthermore, since Alexander polynomial is a topological content of quantum invariants, Alexander polynomial is one of the most important invarinat of knot theory, see [16].…”