Renormalized Reshetikhin-Turaev invariants for the unrolled quantum group of $\mathfrak{sl}_2(\mathbb{C})$

TL;DR

This paper introduces a streamlined approach to the renormalized Reshetikhin-Turaev invariants for links using the category of weight modules over the unrolled quantum group of sl2 at an even root of unity, overcoming limitations of the standard theory.

Contribution

It provides a self-contained, simplified development of the renormalized invariants for the unrolled quantum group of sl2, extending the standard Reshetikhin-Turaev framework.

Findings

Successfully constructs non-trivial link invariants with vanishing quantum dimension components.

Clarifies the relationship between standard and renormalized Reshetikhin-Turaev invariants.

Provides a comprehensive framework applicable to the unrolled quantum group at roots of unity.

Abstract

This paper is a self-contained introduction to the theory of renormalized Reshetikhin-Turaev invariants of links defined by Geer, Patureau-Mirand and Turaev. Whereas the standard Reshetikhin-Turaev theory of a $\mathbb{C}$-linear ribbon category assigns the trivial invariant to any link with a component colored by a simple object of vanishing quantum dimension, the renormalized theory does not. We give a streamlined development of the renormalized Reshetikhin-Turaev theory of links for the category of weight modules over the restricted unrolled quantum group of $\mathfrak{sl}_2(\mathbb{C})$ at an even root of unity.