The Alexander polynomial as quantum invariant of links

Arkiv för Matematik - Tập 53 - Trang 177-202 - 2014
Antonio Sartori1
1Mathematisches Institut, Universität Bonn, Bonn, Germany

Tóm tắt

In these notes we collect some results about finite-dimensional representations of $U_{q}(\mathfrak {gl}(1\mid1))$ 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 $U_{q}(\mathfrak {gl}(1\mid1))$ -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 $U_{q}(\mathfrak {gl}(1\mid1))$ .

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Arkiv för Matematik

Tập 53

177-202

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