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Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An essential surface is an incompressible and $\partial$-incompressible surface.

My question is this: Let $K$ be a Montesinos knot in $S^3$ and $\eta(K)$ is an open neighborhood of $K$. Does every essential surface, that is not isotopic to the boundary, intersect the boundary of $S^3-\eta(K)$?

If $K'$ is a knot such that $S^3-\eta(K')$ is Seifert fibered, this result holds as proved by Zupan in Lemma 3.5

Any references or suggestions are appreciated. Thanks.

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Most Montesinos knots and links have closed incompressible surfaces in their complements. This was shown by Ulrich Oertel in the paper "Closed incompressible surfaces in complements of star links", Pac. J. Math. 111 (1984), 209-230. (Star links are an old name for Montesinos links.) Oertel showed that all closed incompressible surfaces are isotopic to surfaces constructed in a very special way by tubing together a collection of incompressible 4-punctured spheres with meridional boundary, and he gave criteria for when such tubings produce incompressible surfaces. From this one sees that closed incompressible surfaces exist in most cases.

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