Ooguri–Vafa integrality conjecture for the framed unknot

Let $\tau\in\mathbb{Z}$ be a framing, let $Z^{(\mathbb{C}^3,\mathcal{D}_\tau)}(\mathbf{x};q)$ be the framed open-string partition function, and let $f_n^\tau(q)$ be the coefficient of $x^n$ in its plethystic logarithm after multiplication by $q^{1/2}-q^{-1/2}$. Ooguri–Vafa conjecture. For every $\tau\in\mathbb{Z}$ and fixed $n\geq1$, there are integers $N_{n,k}(\tau)$, only finitely many nonzero, such that

$$f_n^\tau(q)=\sum_{k\in\mathbb{Z}}N_{n,k}(\tau)q^{k/2}\in\mathbb{Z}[q^{\pm1/2}].$$

This is the specialization of the open-string integrality prediction to $(\mathbb{C}^3,\mathcal{D}_\tau)$; the paper proves it for $\tau\leq0$ and investigates the remaining framings.