The Orthogonal LMOV conjecture for orthogonal Chern–Simons invariants

Let $\mathcal{L}$ be a link with $L$ components, and let $Z_{CS}^{SO}(\mathcal{L},q,t)$ be its orthogonal Chern–Simons partition function. Write

$$F^{SO}(\mathcal{L},q,t)=\log Z_{CS}^{SO}(\mathcal{L},q,t)=\sum_{\overrightarrow{\mu}\neq\overrightarrow{0}}F_{\overrightarrow{\mu}}^{SO}pb_{\overrightarrow{\mu}}(\overrightarrow{z}),$$

and define

$$g_{\overrightarrow{\mu}}(q,t)=\sum_{k\mid\overrightarrow{\mu}}\frac{\mu(k)}{k}F_{\overrightarrow{\mu}/k}^{SO}(q^k,t^k),$$

where $\mu(k)$ is the Möbius function. Orthogonal LMOV conjecture. For every nonzero tuple of partitions $\overrightarrow{\mu}=(\mu^1,\ldots,\mu^L)$, with $\mathrm{z}_{\overrightarrow{\mu}}$ the associated combinatorial factor,

$$\frac{\mathrm{z}_{\overrightarrow{\mu}}(q-q^{-1})^2\left[g_{\overrightarrow{\mu}}(q,t)-g_{\overrightarrow{\mu}}(q,-t)\right]}{2\displaystyle\prod_{\alpha=1}^{L}\prod_{i=1}^{\ell(\mu^\alpha)}(q^{\mu_i^\alpha}-q^{-\mu_i^\alpha})}\in\mathbb{Z}[q-q^{-1}][t,t^{-1}].$$

This is the orthogonal quantum-group analogue of the LMOV integrality conjecture for colored HOMFLY and Kauffman polynomials. The assertion concerns the integrality of reformulated invariants; its status is not established by the supplied text.