Congruent skein relations for colored Jones polynomials of links

Let $\mathcal{L}$ be a link with $L\geq2$ components, and let $\mathcal{L}_+$, $\mathcal{L}_-$ denote links differing at one crossing between two components. Let $J_N(\mathcal{L};q)$ be the $N$-colored Jones polynomial, and define

$$A_n=\{q\mid q^n=\pm1\},\qquad B_n=\{q\mid q^n=1\},\qquad C_n=\{q\mid q^n=-1\}.$$

Congruent skein relations. For any such link,

$$J_N(\mathcal{L}_+;q)-J_N(\mathcal{L}_-;q)\equiv0\pmod{[N]},$$

and

$$J_N(\mathcal{L}_+;q)-J_N(\mathcal{L}_-;q)\equiv0\pmod{[N+2]}.$$

Moreover, if $L$ is odd with $L\geq3$ and $N>k\geq1$, the roots of $J_N(\mathcal{L}_+;q)-J_N(\mathcal{L}_-;q)=J_k(\mathcal{L}_+;q)-J_k(\mathcal{L}_-;q)$ contain $(A_{N-k}\cup A_{N+k+2})-(A_{k+1}-A_1)$. If $L$ is even, the corresponding root sets contain $(B_{N-k}\cup C_{N+k+2})-(A_{k+1}-A_1)$ and $(C_{N-k}\cup B_{N+k+2})-(A_{k+1}-A_1)$ for the two displayed differences in the source. These proposed congruences extend colored-Jones skein phenomena from knots to links; the source presents them after testing examples and does not give a proof of the full statement.