The alternating-knot characterization conjecture via colored Jones degrees

Let $K\subset S^3$ be a knot, let $p_K$ denote its Jones period, and let $d_+[J_K(n)]$ and $d_-[J_K(n)]$ denote the maximal and minimal degrees of its colored Jones polynomial in $t$. Alternating-knot characterization conjecture. The knot $K$ is alternating if and only if

$$p_K=1\quad\text{and}\quad 2d_+[J_K(n)]-2d_-[J_K(n)]=cn^2+(2-c)n-2$$

for some $c\in\mathbb{Z}$. This proposes detecting alternating knots from the Jones period and degree span. The forward direction is standard from the alternating-knot formulas, while the converse is supported by the strong slope conjecture and results on Turaev genus; the conjecture itself is not resolved in the supplied text.