The Alexander polynomial as quantum invariant of links

Antonio Sartori Mathematisches Institut, Universität Bonn

Quantum Algebra Representation Theory mathscidoc:1701.29004

Arkiv for Matematik, 53, (1), 177-202, 2013.8
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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@inproceedings{antonio2013the,
  title={The Alexander polynomial as quantum invariant of links},
  author={Antonio Sartori},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639181852072},
  booktitle={Arkiv for Matematik},
  volume={53},
  number={1},
  pages={177-202},
  year={2013},
}
Antonio Sartori. The Alexander polynomial as quantum invariant of links. 2013. Vol. 53. In Arkiv for Matematik. pp.177-202. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639181852072.
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