The residue formula for almost abelian 0-surgery invariants

Let $K_n$ be the twist knot indexed by $n$, let $F_{K_n}(x,q)$ be its two-variable invariant, and let $\Delta_{K_n}(x)=nx+nx^{-1}-(2n-1)$ be its Alexander polynomial. Choose a solution $x_0$ of

$$\Delta_{K_n}(x)=0.$$

The residue formula.

$$Z_0^{(-)}\left(S^3_0(K_n)\right)\cong\operatorname{Res}_{x=x_0}\frac{x^{1/2}-x^{-1/2}}{x}F_{K_n}(x,q).$$

The two possible choices of $x_0$ give residues differing only by sign. This conjecture proposes that the almost abelian invariant for twist-knot 0-surgery is recovered from a residue of the two-variable series at a root of the Alexander polynomial, suggesting a link with the unipotent branch of the flat-connection moduli space.