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April 2015 The Alexander polynomial as quantum invariant of links
Antonio Sartori
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Ark. Mat. 53(1): 177-202 (April 2015). DOI: 10.1007/s11512-014-0196-5


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 $U_{q}(\mathfrak {gl}(1\mid1))$.

Funding Statement

This work has been supported by the Graduiertenkolleg 1150, funded by the Deutsche Forschungsgemeinschaft.


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Antonio Sartori. "The Alexander polynomial as quantum invariant of links." Ark. Mat. 53 (1) 177 - 202, April 2015.


Received: 9 August 2013; Revised: 20 December 2013; Published: April 2015
First available in Project Euclid: 30 January 2017

zbMATH: 1329.57021
MathSciNet: MR3319619
Digital Object Identifier: 10.1007/s11512-014-0196-5

Rights: 2014 © Institut Mittag-Leffler


Vol.53 • No. 1 • April 2015
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