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

Let ${\cal L}$ be an unoriented link with $L$ components, colored by representations $R_1,\ldots,R_L$. Let $\hat f_{R_1,\ldots,R_L}(t,\nu)$ denote the reformulated colored HOMFLY invariant obtained from the link invariants using the inverse matrices associated with each coloring, and set $z=t-t^{-1}$. Labastida–Mariño–Ooguri–Vafa link integrality conjecture. One has

$$\hat f_{R_1,\ldots,R_L}(t,\nu)\in z^{L-2}{\mathbb Z}[z^2,\nu^{\pm1}],$$

that is,

$$\hat f_{R_1,\ldots,R_L}(t,\nu)=z^{L-2}\sum_{g\geq 0}\sum_{Q\in{\mathbb Z}}N_{R_1,\ldots,R_L;g,Q}z^{2g}\nu^Q.$$

This extends the predicted BPS integrality structure from knots to links and is part of the general integrality framework for colored HOMFLY invariants.