Dimofte–Garoufalidis 1-loop conjecture for hyperbolic knot complements

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated with the complete hyperbolic structure. Let $X$ be a $\rho_0$-regular ideal triangulation with a shape solution $\mathbf z_0$ representing $[\rho_0]$, let $\mathcal{V}_0(X)$ be the irreducible component of the gluing variety containing $\mathbf z_0$, and let $\tau$ and $\mathbb T$ denote the 1-loop invariant and adjoint twisted Reidemeister torsion. Dimofte–Garoufalidis 1-loop conjecture. For every $\mathbf z\in\mathcal{V}_0(X)$ with ${\mathcal P}_X(\mathbf z)=[\rho_{\mathbf z}]$,

$$\tau({\mathbb S}^3{\smallsetminus} K,\boldsymbol\gamma,X,\mathbf z)=\pm\mathbb T_{({\mathbb S}^3{\smallsetminus} K,\boldsymbol\gamma)}([\rho_{\mathbf z}]).$$

The conjecture identifies the state-integral 1-loop quantity with Reidemeister torsion and is used here as the conceptual background for the paper's 1-loop asymptotics; no resolution of the full statement is supplied.