Congruent skein relations for reformulated colored HOMFLY-PT invariants

Let $\mathcal{L}$ be a link and let $p$ be a prime number. Write $\check{\mathcal{Z}}_p(\mathcal{L};q,t)$ for the reformulated colored HOMFLY-PT invariant associated with the $p$-row coloring. For a crossing, let $\mathcal{L}_+$, $\mathcal{L}_-$, and $\mathcal{L}_0$ denote the positive crossing, negative crossing, and oriented smoothing, respectively. Define $[p]=q^p-q^{-p}$ and $\{p\}=(q^p-q^{-p})/(q-q^{-1})$, and interpret $A\equiv B\pmod C$ as $(A-B)/C\in\mathbb{Z}[(q-q^{-1})^2,t^{\pm1}]$. Congruent skein relations. For any link $\mathcal{L}$ and prime $p$, if the crossing is a self-crossing of a knot, then

$$\check{\mathcal{Z}}_{p}(\mathcal{L}_+;q,t)-\check{\mathcal{Z}}_{p}(\mathcal{L}_-;q,t)\equiv (-1)^{p-1}\check{\mathcal{Z}}_{p}(\mathcal{L}_0;q,t)\pmod{\{p\}^2}.$$

If the crossing links two different components, then

$$\check{\mathcal{Z}}_{p}(\mathcal{L}_+;q,t)-\check{\mathcal{Z}}_{p}(\mathcal{L}_-;q,t)\equiv (-1)^{p-1}p[p]^2\check{\mathcal{Z}}_{p}(\mathcal{L}_0;q,t)\pmod{\{p\}^2[p]^2}.$$

These relations are proposed as a higher-color analogue of the classical HOMFLY-PT skein relation and are motivated by the LMOV conjecture; the source does not establish their general validity.