Non-semisimple invariants and Habiro’s series

Abstract

In this chapter 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 results to hold for all Seifert genus 1 knots.