3d-3d correspondence and abelian flat connection

We realize a homological block of a knot complement in $S^3$ for $G_{\mathbb{C}}=SL(2,\mathbb{C})$ as a half-index of a 3d $\mathcal{N}=2$ theory via an expression of the homological block as an inverted Habiro series by working out some examples, which we expect to extend to general knots.

Also, by choosing a certain set of poles in the integral expression of the half-index, we obtain the colored Jones polynomial.