Relations between Reshetikhin--Turaev and re-normalized link invariants

TL;DR

This paper explores the relationship between Reshetikhin--Turaev and re-normalized link invariants, revealing that Reshetikhin--Turaev invariants appear in higher order terms for plumbed links, extending prior residue-based relations.

Contribution

It demonstrates that Reshetikhin--Turaev invariants are present in higher order terms of re-normalized invariants for plumbed links, beyond residue relations.

Findings

Reshetikhin--Turaev invariants appear in higher order terms

Residue relations do not hold for links, only for knots

Extension of relations to plumbed links

Abstract

Costantino--Geer--Patureau-Mirand proved relations between the Reshetikhin--Turaev link invariants and the re-normalized link invariants for knots. Their theorem says that residues of the re-normalized link invariants are given by the Reshetikhin--Turaev link invariants. However, in the case of links, the residues are vanish, so we can not obtain any relations from the residues. In this paper, we prove that the Reshetikhin--Turaev link invariants appear in higher order terms of the re-normalized link invariants for links which are plumbed.