Non-semisimple invariants and Habiro's series (Q2055080)

Let \(J_K(q^N,q)\) denote the \(N\)-colored Jones polynomial of a knot \(K\) in the \(3\)-sphere. \textit{K. Habiro} [Invent. Math. 171(1), 1--81 (2008; Zbl 1144.57006)] introduced the cyclotomic expansion (Habiro's series) \(J_K(x,q) = \sum_{m\geq 0} a_m(K;q)\sigma_m(x,q)\) in the cyclotomic completion of the center of \(U_q(\mathfrak{sl}_2)\), where \(\sigma_m(x,q)\) is a certain Laurent polynomial in \(x\) and \(q\). In the article under review, the authors connect the cyclotomic expansion with non-semisimple invariants which arise in specializations of the quantum \(\mathfrak{sl}_2\) at \(q=e_p\), where \(e_p\) is a primitive \(p\)th root of unity. More precisely, they discuss the Akutsu-Deguchi-Ohtsuki (ADO) invariant for knots, the Witten-Reshetikhin-Turaev (WRT) invariant and Costantino-Geer-Patureau (CGP) invariant for 3-manifolds. First, the authors reformulate a result of \textit{S. Willetts} [Quantum Topol. 13, No. 1, 137--181 (2022; Zbl 1494.57019)] and prove Theorem~2.1: \(J_K(x,e_p) = \mathrm{ADO}_K(x,e_p)/\Delta_K(x^p)\), where \(\Delta_K\) denotes the Alexander polynomial of \(K\). Here they focus on a double twist knot \(K\) and show Theorem~2.2: \(\mathrm{ADO}_K(x,e_p) = \sum_{n=0}^{p-1} a_n(K;e_p)\sigma_n(x,e_p)\). Next, the authors turn their attention to the \(3\)-manifold \(M\) obtained from \(S^3\) by \(0\)-surgery on a double twist knot \(K\). In Theorem~2.3, they prove that the CGP invariant of \(M\) is expressed by the WRT invariant of \(M\) and the coefficient \(a_{p-1}(K;e_p)\). As a consequence, \(a_{p-1}(K;e_p)\) is a topological invariant of \(M\). Here they mention that ``it is an interesting open problem to find a topological interpretation of this invariant'' and propose a conjecture: Theorems~2.2 and 2.3 hold for any Seifert genus \(1\) knot. Also, in Theorem~2.5, the authors give a formula for the ADO invariant of the \((2,2t+1)\)-torus knot with \(t\geq 2\). For the entire collection see [Zbl 1466.57001].