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There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot.

I am curious about the other direction:

If $K$ is a fibered knot which is a closure of a braid $\beta$, then what can we say about $\beta$? Is there any structural theorem of fibered knots in terms of their braid words?

Any comment or suggestion about references will be greatly appreciated.

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    $\begingroup$ I think this might be of interest: arxiv.org/pdf/2004.07445.pdf Section 7 has a discussion along the lines you are asking about. $\endgroup$
    – Neil Hoffman
    Apr 20, 2021 at 2:11
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    $\begingroup$ Note that $\beta$ is fibered with fiber surface equal to the surface given by Seifert's algorithm applied to $\beta$ if and only if $\beta$ is homogeneous. Also, this might interest you: arxiv.org/abs/1610.09664 $\endgroup$
    – Lukas Lewark
    Apr 20, 2021 at 13:00

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