Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries (Q6655623)

Let \(M\) be a connected compact oriented manifold and \(\mathfrak{g}\) denote the Lie algebra of a semisimple complex Lie group \(G\). Define \(R^{irr}_G(M)\) as the set of conjugacy classes of irreducible representations \(\pi_1(M)\rightarrow G\) which is called the irreducible character variety. The adjoint Reidemeister torsion with coefficients twisted by a representation \(\varphi:\pi_1(M)\rightarrow G\), denoted by \(\tau_\varphi(M)\in \mathbb{C}^\times\), is defined to be the alternative product of determinants under a mild assumption. The author focuses on the adjoint Reidemeister torsion of \(3\)-manifolds \(M\) with no boundary.\N\NFollowing [\textit{F. Benini} et al., J. High Energy Phys. 2020, No. 3, Paper No. 57, 40 p. (2020; Zbl 1435.83071) and \textit{D. Gang} et al., ibid. 2020, No. 3, Paper No. 164, 43 p. (2020; Zbl 1435.81161)] , the author considers the following Conjecture 1.1: The sum of the \(n\)th powers of the twice torsions \(\sum_{\varphi \in R^{irr}_G(M)}(2\tau_\varphi(M))^n\) lies in the ring of integers \(\mathbb{Z}\) for a closed \(3\)-manifold \(M\). Here, the set \(R^{irr}_G(M)\) is finite and \(n \in \mathbb{Z}\) with \(n\geq -1.\) Moreover, when \(G=SL_2(\mathbb{C)},\) \(M\) is a hyperbolic \(3\)-manifold, and \(n=-1,\) the sum is zero. \N\NLet \(S^3_{p/q} (K)\) be the closed \(3\)-manifold obtained by \((p/q)\)-Dehn surgery on a knot \(K\) in \(S^3\) for \(p/q \in \mathbb{Q}\). The author shows that, for any integers \(p\) and \(q\neq 0\), the vanishing Conjecture 1.1 is true when \(M=S^3_{p/1}(4_1)\) and \(M=S^3_{1/q} (4_1)\), where \(G=SL_2(\mathbb{C)},\) \(4_1\) is the figure-eight knot, and \(n=-1\). In the final section, the author also proves that Conjecture~1.1 is true for \(M = S^3_{1/q}(K)\) when \(K\) is the \(5_2\)-knot.