Adequacy characterization by Jones polynomial span and Turaev genus

Let $L$ be a non-split link, with crossing number $c(L)$, Jones polynomial $V_L(t)$, and Turaev genus $g_T(L)$. A link is adequate when it has a diagram whose all-$A$ and all-$B$ states have no self-intersections.

Adequacy–span conjecture. A non-split link is adequate if and only if

$$\operatorname{span} V_L(t)=c(L)-g_T(L).$$

The theorem proved in the paper establishes this characterization when the Turaev genus is one. The conjecture proposes the corresponding statement for links of arbitrary Turaev genus.