2007/11/30 by Nathan Geer, Bertrand Patureau-Mirand, Vladimir Turaev · 1 citation
Mathematics · #math.QA #math.GT
paper · pdf · doi:10.1112/s0010437x08003795
published as Compositio Mathematica, volume 145 (2009), issue 01, pp. 196-212 · 16 pages
arxiv created 2013/09/25 · arxiv updated 2013/09/26
In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.