Labastida–Mariño–Ooguri–Vafa integrality conjecture for colored link invariants

Let $\mathcal{L}$ be a link with $L$ components, let $\mathcal{P}$ be the set of all partitions, and write $\vec{A}=(A^{1},\ldots,A^{L})\in\mathcal{P}^{L}$. Let $f_{\vec{A}}(q,t)$ be the coefficients in the plethystic expansion of the Chern–Simons free energy. For any $A,B\in\mathcal{P}$, define

$$M_{AB}(q)=\sum_{\mu}\frac{\chi_A(C_\mu)\chi_B(C_\mu)}{\mathfrak{z}_\mu}\prod_{j=1}^{\ell(\mu)}(q^{-\mu_j/2}-q^{\mu_j/2}).$$

Labastida–Mariño–Ooguri–Vafa conjecture. For any $\vec{A}\in\mathcal{P}^{L}$, there exist $P_{\vec{B}}(q,t)$ for $\vec{B}\in\mathcal{P}^{L}$ such that

$$f_{\vec{A}}(q,t)=\sum_{|B^\alpha|=|A^\alpha|}P_{\vec{B}}(q,t)\prod_{\alpha=1}^{L}M_{A^\alpha B^\alpha}(q).$$

Furthermore,

$$P_{\vec{B}}(q,t)=\sum_{g=0}^{\infty}\sum_{Q\in\mathbb{Z}/2}N_{\vec{B};\,g,Q}(q^{-1/2}-q^{1/2})^{2g-2}t^{Q},$$

and $N_{\vec{B};\,g,Q}$ are integers.

This conjecture predicts a nontrivial reformulation of the Chern–Simons free energy in terms of integer invariants, reflecting the duality between Chern–Simons gauge theory and topological string theory. The source does not provide evidence resolving the claim.