Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus. One can prove that, if $S \setminus K$ is connected, then it's essential in the knot complement $M=\mathbb{S}^3 \setminus K$. On the other hand, when not, then it provides two copies of a Seifert surface of minimal genus. As it has no reason to be connected, i wonder if one can determine precisely when it is connected.

Similarly, take a checkerboard surface of a diagram $D$ of $K$, then it's a theorem from Ozawa that it's essential in $M$ iff $D$ is $\textit{semi-adequate}$. But it has no reason to be oriented, and again there is two cases : if it is, then it provides a Seifert surface, and gluing two copies of it along $K$ will give a surface $S$ as in the first paragraph (is it of minimal genus?), with $K \subset S$ separating. If not, then the boundary of its orientation covering will be an essential, oriented surface with two boundary components, and again gluing those components together will give an $S$ as in the first paragraph, with $K \subset S$ non-separating.

Are those two constructions equivalent? What happens for non semi-adequates knots? Is there a characterization of knots for which checkerboard surfaces are oriented or not?

Thanks for any answer or comment.