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.

  • $\begingroup$ 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? $\endgroup$ Mar 21, 2016 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, 2016 at 23:35

1 Answer 1


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.

  • $\begingroup$ 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, 2016 at 18:22
  • 1
    $\begingroup$ @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, 2016 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? $\endgroup$ Mar 23, 2016 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, 2016 at 16:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.