Given a fibered knot $K\subseteq S^3$, does every genus-minimizing Seifert surface appear as the fiber of a bundle $S^3\setminus K\to S^1$?
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$\begingroup$ I think your question is answered here: kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2263-04.pdf see Sec. 4. $\endgroup$– Ryan BudneyCommented Aug 26 at 22:56
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The answer is "yes". See Corollary 2 on page 119 of Thurston's article A norm for the homology of 3-manifolds.
In fact, fibered knots fibre in exactly one way. And all minimal genus Seifert surfaces (for a fixed fibered knot) are isotopic.
Since Thurston's paper is not immediately and obviously available online (?), I googled around and found A survey of the Thurston norm. See Example 2.4 therein.
