The LMOV conjecture for a framed knot

Let $\mathcal{K}_{\tau}$ be a framed knot with $\tau\in\mathbb{Z}$, and let $f_m(\mathcal{K}_{\tau};q,a)$ be the reduced free-energy coefficients defined from the reduced Chern–Simons partition function. Reduced LMOV conjecture for framed knots. There exist integers $N_{m,i,k}(\tau)$, with only finitely many nonzero for each fixed $m\geq1$, such that

$$f_m(\mathcal{K}_{\tau};q,a)=-\sum_{i,k\in\mathbb{Z}}\frac{N_{m,i,k}(\tau)a^{i/2}q^{(k+1)/2}}{1-q}.$$

This is the reduced, weak integrality form of the framed-knot LMOV conjecture; the supplied text gives no resolution status.