Labastida–Mariño–Vafa integrality conjecture for reformulated colored HOMFLY polynomials

Let $\mathcal{L}$ be an oriented link with $l$ components. For partitions $\lambda^1,\dots,\lambda^l$, define the reformulated colored HOMFLY polynomials $f_{\lambda^1,\dots,\lambda^l}(t,\nu)$ through the plethystic expansion of the logarithm of the generating function of the colored HOMFLY polynomials. For $\lambda,\mu\vdash n$, set

$$M_{\lambda\mu}(t)=\sum_{\tau\vdash n}\frac{|C_\tau|}{n!}\chi^\lambda(C_\tau)\chi^\mu(C_\tau)\frac{\prod_{j=1}^{\ell(\tau)}(t^{-\tau_j/2}-t^{\tau_j/2})}{t^{-1/2}-t^{1/2}}.$$

Labastida–Mariño–Vafa conjecture. For partitions $\lambda^1,\dots,\lambda^l$,

$$f_{\lambda^1,\dots,\lambda^l}(t,\nu)=\sum_{\mu^1\vdash|\lambda^1|,\dots,\mu^l\vdash|\lambda^l|}\widehat f_{\mu^1,\dots,\mu^l}(t,\nu)\,M_{\lambda^1\mu^1}(t)\cdots M_{\lambda^l\mu^l}(t),$$

where

$$\widehat f_{\mu^1,\dots,\mu^l}(t,\nu)=\sum_{g\geq0}\sum_Q N_{\mu^1,\dots,\mu^l,g,Q}(t^{1/2}-t^{-1/2})^{2g+l-2}\nu^Q,$$

with $N_{\mu^1,\dots,\mu^l,g,Q}$ integers and $Q$ either all integers or all semi-integers. The integers $N_{\mu^1,\dots,\mu^l,g,Q}$ are interpreted as quantities involved in the enumerative geometry of the resolved conifold. The conjecture had been verified only for trivial links and some simplest knots and links with small partitions; the paper verifies it for several infinite families of torus links with small partitions, but a general proof remains open.