Irreducible knot-state asymptotics

Let $K$ be a knot with exterior $E_K$ and peripheral torus $\Sigma$. Let ${\mathcal M}^s(\Sigma)$ and ${\mathcal M}^s(E_K)$ denote the smooth loci of the corresponding character moduli spaces, and let $r:{\mathcal M}(E_K)\to{\mathcal M}(\Sigma)$ be restriction. For an open set $U\subset{\mathcal M}^s(\Sigma)$, set $V=r^{-1}(U)$ and assume that $V$ is connected, contained in ${\mathcal M}^s(E_K)$, and that $r|_V$ is an embedding. Irreducible knot-state asymptotics conjecture. On $U$,

$$Z_k(E_K)=e^{i m\pi/4}\frac{k^{3/4}}{4\pi^{3/4}}F^k f(\mathord\cdot,k)+O(k^{-\infty}),$$

where $m$ is an integer, $F$ is a section of $L\to U$ satisfying $F(r(\rho))=\operatorname{CS}(\rho)$ for $\rho\in V$ and the Cauchy–Riemann equation up to a term vanishing to infinite order along $r({\mathcal M}^s)$, and $f(\mathord\cdot,k)$ has an asymptotic expansion $f_0+k^{-1}f_1+\cdots$ with coefficients in ${\mathcal C}^{\infty}(U,\delta)$ satisfying $\bigl(r^*\varphi(f_0^2)\bigr)(\rho)=\pm\mathbb T(\rho)$. The conjecture gives the expected Chern–Simons phase and Reidemeister-torsion symbol near regular irreducible representations; the supplied text does not state a general resolution.