Non-semisimple invariants and Habiro's series

TL;DR

This paper links Habiro's cyclotomic expansion of the colored Jones polynomial at roots of unity with ADO invariants of double twist knots, enabling comparison of WRT and CGP 3-manifold invariants and revealing their differences.

Contribution

It establishes an explicit relationship between Habiro's series and ADO invariants for double twist knots, facilitating comparison of different quantum invariants.

Findings

The difference between WRT and CGP invariants is given by the p-1 coefficient of Habiro's series.

The relationship is explicitly established for double twist knots.

Results are expected to extend to all Seifert genus 1 knots.

Abstract

In this paper we establish an explicit relationship between Habiro's cyclotomic expansion of the colored Jones polynomial (evaluated at a p-th root of unity) and the Akutsu-Deguchi-Ohtsuki (ADO) invariants of the double twist knots. This allows us to compare the Witten-Reshetikhin-Turaev (WRT) and Costantino-Geer-Patureau (CGP) invariants of 3-manifolds obtained by 0-surgery on these knots. The difference between them is determined by the p-1 coefficient of the Habiro series. We expect these to hold for all Seifert genus 1 knots.