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We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber $\overline{\Sigma}$. Given a fibered knot $K$ of genus $g$, by gluing two copies of $\overline{\Sigma}$ together we get a Heegaard surface for $M$ of genus $2g$.

Is it known whether the difference between the Heegaard genus of $M$ and the minimum of twice the genus of a fibered knot in $M$ can be arbitrarily large?

In general, are there techniques besides Heegaard genus to bound from below the minimal genus of a fibered knot in a 3-manifold? I know that Ken Baker has worked on counting genus one fibered knots in lens spaces, but I don't know of any other work in this area.

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  • $\begingroup$ I am confused by your setup here. If $M\cong S^3$, then can't we just choose a sequence of fibered knots of increasing and unbounded genus to answer your first question positively? $\endgroup$
    – Neil Hoffman
    Mar 21 '16 at 23:32
  • $\begingroup$ @NeilHoffman Thanks for the comment, see the edit. I want to minimise the genus over all fibered knots. $\endgroup$
    – magicker72
    Mar 21 '16 at 23:35
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This difference can be made arbitrarily large, so that there is a manifold of Heegaard genus $g$ such that the minimum genus of a fibered knot is at least $g$. This follows from examples of Hass-Thompson-Thurston of closed orientable hyperbolic 3-manifolds of Heegaard genus $g$ which have a genus $g$ oriented Heegaard splitting which requires $g$ stabilizations to become isotopic to the stabilization of the Heegaard splitting with the opposite orientation.

Now, suppose one has such a manifold of Heegaard genus $g$ which has curve complex distance $\geq 2g$ (which is essentially a corollory of the manner in which the manifolds are constructed). Suppose one has a fibered knot of genus $<g$. The corresponding Heegaard splitting of genus $<2g$ has the property that it is a stabilization of the Heegaard splitting of genus $g$ by a Corollary 4.5 of Scharlemann and Tomova. However, the Heegaard splitting of a fibered knot is equivalent to its orientation reversal, a contradiction.

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  • $\begingroup$ Thanks! One question: in the second paragraph, don't we need curve complex distance $>2g$ in order to apply Corollary 4.5 of Scharlemann--Tomova? $\endgroup$
    – magicker72
    Mar 22 '16 at 18:22
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    $\begingroup$ @magicker72: okay, correct, I fixed that. In any case, one can make the distance arbitrarily large by taking powers of the gluing map. One remark: most likely there are much simpler examples, namely lens spaces, but I wasn't sure if this is known (Ken might know a reference). $\endgroup$
    – Ian Agol
    Mar 22 '16 at 20:14
  • $\begingroup$ There are some complications for this problem when considering knots in lens spaces, since every lens space admits a genus 0 fibered knot, the standard unknot. However, excluding knots of this type (say by placing restrictions a homotopy or homology class of the knot) would be a natural place to look for the type of simpler examples of knots in lens spaces you mention. Alternatively, perhaps one could consider small Seifert fibered spaces as a means of avoiding the unknot surgeries? $\endgroup$
    – Neil Hoffman
    Mar 23 '16 at 1:36
  • $\begingroup$ @NeilHoffman My definition of fibered is integrally fibered, ie. that the fiber surfaces are honest Seifert surfaces, not rational ones. The standard "unknot" in $L(p, q)$ for $p > 1$ is rationally, not integrally, fibered. $\endgroup$
    – magicker72
    Mar 24 '16 at 16:45

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