  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.
