LMOV integrality conjecture for colored HOMFLY‑PT invariants

Prove that the numbers N_{r,i,j} appearing in the expansion of the colored HOMFLY‑PT generating series P(K)(x,a,q) = exp(∑_{n,r≥1} (1/n) f_r(a^n, q^n) x^{rn}) with f_r(a,q) = ∑_{i,j} N_{r,i,j} a^i q^j / (q − q^{−1}) are integers for all knots.

Background

The LMOV conjecture predicts integrality of BPS numbers derived from colored HOMFLY‑PT generating functions. The paper reviews the standard re‑expansion of P(K) that defines N_{r,i,j} and recalls that, for knots admitting a (generalized) knots–quivers correspondence, these integers follow from properties of motivic Donaldson–Thomas invariants of symmetric quivers.

While the authors show integrality for the class of knots covered by the generalized correspondence via a proposition, the general conjecture remains open and is explicitly stated in its standard form.