The arc-index–Turaev-genus inequality for non-split prime links

Let $L$ be a non-split prime link. Its crossing number is $c(L)$, its arc index is $b1(L)$, and its Turaev genus is $g_T(L)$.

Arc-index–Turaev-genus conjecture.

$$c(L) + 2 - \alpha(L) \geq 2g_T(L).$$

This conjecture relates the deficit of the arc index from the alternating-link value to the Turaev genus, measuring how far a link is from being alternating. It is verified in the paper for adequate links and for Kanenobu knots outside a specified exceptional family, but is not resolved in general.