Modified quantum dimensions and re-normalized link invariants

TL;DR

This paper introduces a re-normalization technique for quantum link invariants using modified quantum dimensions, enabling non-trivial invariants in cases where traditional quantum dimensions vanish, with applications to Lie superalgebras and nilpotent representations.

Contribution

It presents a new re-normalization method for quantum invariants that produces non-zero results where usual quantum dimensions vanish, expanding the scope of link invariants.

Findings

Modified quantum dimensions can be non-zero even when usual quantum dimensions vanish.

New link invariants derived from Lie superalgebras generalize the multivariable Alexander polynomial.

Hierarchy of invariants from nilpotent representations includes Kashaev's invariants.

Abstract

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.