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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.

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  • $\begingroup$ I don't understand your first paragraph. What do you mean by "this case"? $\endgroup$ – Ryan Budney Jun 1 '16 at 21:23
  • $\begingroup$ Edited : "...when it's connected" $\endgroup$ – Léo Jun 2 '16 at 9:28
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It is not true in general that if $S\setminus K$ is connected, then it is essential, even if $S$ is assumed to have minimal genus, since every knot embeds in a torus $S$ such that $S\setminus K$ is a boundary parallel annulus.

In fact, the Neuwirth conjecture, which is still open, asks whether every non-trivial knot in $S^3$ can be embedded in a closed surface $S$ such that $S\setminus K$ is both connected and essential in $M$. A stronger version of the conjecture asked whether every non-trivial non-torus knot bounds an essential non-orientable spanning surface, but counterexamples to this statement were found by Dunfield. Although a weaker version, sometimes also called the Neuwirth conjecture, was proven by Culler and Shalen.

Another way to see that the two constructions are not equivalent, is to notice that for homological reasons any spanning surface must have even integral boundary slope, whereas it is possible for $S\setminus K$ to have odd boundary slope.

A checkerboard surface is orientable if and only if its Tait graph is bipartite, and every knot admits a diagram with this property.

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