Labastida–Mariño–Ooguri–Vafa integrality conjecture for colored HOMFLY invariants of knots

Let ${\cal K}$ be a knot, let $R$ be a representation, and let $\hat f_R(t,\nu)$ be the reformulated colored HOMFLY invariant obtained by applying the inverse matrix $M^{-1}$ to the reformulated invariant $f_R(t,\nu)$. Set $z=t-t^{-1}$. The integers $N_{R;g,Q}$ are indexed by $g\geq 0$ and $Q\in{\mathbb Z}$. Labastida–Mariño–Ooguri–Vafa integrality conjecture. The reformulated invariant satisfies

$$\hat f_R(t,\nu)\in z^{-1}{\mathbb Z}[z^2,\nu^{\pm1}],$$

or equivalently

$$\hat f_R(t,\nu)=z^{-1}\sum_{g\geq 0}\sum_{Q\in{\mathbb Z}}N_{R;g,Q}z^{2g}\nu^Q,$$

where the $N_{R;g,Q}$ are integers and only finitely many are nonzero for fixed ${\cal K}$ and $R$. This is the predicted BPS integrality structure for colored HOMFLY invariants; the source notes that the corresponding conjecture had already been proved.