Abelian knot-state asymptotics

Let $K$ be a knot with exterior $E_K$, let $E=H_1(\Sigma,\mathbb R)$ for the peripheral torus $\Sigma$, and let $\lambda$ be the longitude direction. Let a regular point of $\mathbb R\lambda$ mean a point satisfying the source's condition that a neighborhood of its image avoids irreducible representations and that the Alexander polynomial does not vanish at the corresponding point. Abelian knot-state asymptotics conjecture. Every regular point of $\mathbb R\lambda$ has an open neighborhood $V\subset E$ such that $V\cap\mathbb R\lambda$ consists of regular points and

$$Z_k(E_K)(x)=e^{i m\pi/4}\left(\frac{k}{2\pi}\right)^{1/4}t_\lambda^k(x)\otimes f(x,k)\Omega_\lambda+O(k^{-\infty}),\qquad x\in V,$$

where $m$ is an integer, $t_\lambda$ is the holomorphic section of $L\to E$ restricting to $1$ on $\mathbb R\lambda$, $f(\mathord\cdot,k)$ has an asymptotic expansion $f_0+k^{-1}f_1+\cdots$,

$$f_0(q\lambda)=\frac{1}{\sqrt2}\frac{\sigma-\sigma^{-1}}{\Delta_K(\sigma^2)},\qquad \sigma=e^{2\pi i q},$$

and $\Omega_\lambda\in\delta$ satisfies $\Omega_\lambda^2(\lambda)=1$. This conjecture describes the leading asymptotics near regular abelian representations; the supplied text gives no resolution status.