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Consider a knot $K \subset S^3$ and let $M_K$ be the associated double branched cover. The pre-image $S$ of a Seifert surface is a surface without boundary inside $M_K$.

Can $S$ be incompressible? If yes, how is this related to the topology of $K$?

I'm particularly interested in the case where $M_K$ is hyperbolic.

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The incompressible Seifert surface $\Sigma$ will have incompressible preimage $S$ in the double branched cover if and only if the complement of a tubular neighborhood of $\Sigma$ in $S^3$ has incompressible boundary. This condition was termed "partially unknotted" by Jaco.

Examples of Seifert surfaces which are not partially unknotted were claimed by Jaco in the paper, but not explicitly described. The simplest examples can be obtained by taking a knotted handlebody with hyperbolic complement, such as Thurston's knotted wye.

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See also Adams-Reid for descriptions of knotted genus 2 handlebodies (with incompressible hyperbolic complements).

Now, draw a knot on this handlebody which bounds a genus 1 Seifert surface $\Sigma$, using the fact that for a genus one surface $\Sigma$, $\Sigma\times [0,1]$ is a genus 2 handlebody. Any such knot will have a Seifert surface which is not partially unknotted.

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