Taut foliations and nearly-fibered knots

Problem: Prove that nearly-fibered knots which aren't cables are persistently foliar.

A knot $K\subset S^3$ is said to be persistently foliar if for every rational $r$ the complement $S^3\setminus n(K)$ admits a coorientable taut foliation (CTF) which meets the boundary torus $\partial n(K)$ transversely in a foliation by curves of slope $r$. Note that if a knot $K$ is persistently foliar than every nontrivial surgery on $K$ admits a CTF. The L-space and cabling conjectures together predict that if $K$ is neither a cable nor an L-space knot then every nontrivial surgery on $K$ admits a CTF. It is therefore reasonable to conjecture that such knots are also persistently foliar. This is what Delman-Roberts call the L-space knot conjecture. In particular, L-space knots are fibered. So one expects that every non-fibered, non-cable knot is persistently foliar.

Li-Roberts proved that for any knot there is an interval around 0 such that for every slope in the interval there is a CTF in the knot complement foliating the boundary in curves of that slope. They construct these foliations using branched surfaces in the knot complement assembled from the Seifert surface for the knot together with surfaces appearing in a taut sutured manifold heirarchy of the Seifert surface complement. One expects that this interval should be at least $(-1,1)$ for every knot (this is also the largest interval that can be realized in general; this would show that every toroidal homology sphere admits a taut foliation, completing the proof of the L-space conjecture for such manifolds), but not knowing specifics about the surfaces in the hierarchy makes it hard to control the size of the interval. The stated problem is to show that one can realize every nontrivial slope for non-cabled nearly-fibered knots, the idea being that we understand their Seifert surface complements very concretely.

For background: a knot $K\subset S^3$ is said to be nearly-fibered if its knot Floer homology satisfies $$\mathrm{dim}\widehat{\mathit{HFK}}(K,g) = 2.$$ Recall that $K$ is fibered if and only if $$\mathrm{dim}\widehat{\mathit{HFK}}(K,g) = 1,$$ by work of Ni and Ghiggini, so nearly-fibered knots are those non-fibered knots that are as close as possible, from the point of view of Floer homology, to being fibered. This notion was introduced by Baldwin-Sivek who then completely classified the genus-1 nearly fibered knots in $S^3$. They are $$5_2, \,\,\, 15_{n43522}, \,\,\, \mathrm{Wh}^+(T_{2,3};2), \,\,\, \mathrm{Wh}^-(T_{2,3};2), \,\,\, P(-3,3,2n+1),$$ where $\mathrm{Wh}^\pm(T_{2,3};2)$ are the positively and negatively-clasped 2-twisted Whitehead doubles of the trefoil, and the $P(-3,3,2n+1)$ are pretzel knots. Presumably these are persistently foliar but one should check!

Although defined originally in terms of Floer homology, Li-Ye showed that nearly-fibered knots admit a purely topological description in terms of the sutured manifold decompositions of their Seifert surface complements. As hinted above, their concrete description of these guts should allow one to build laminar branched surfaces a la Li-Roberts with product sutured complementary regions in nearly-fibered knot complements, with more control over the boundary slopes realized by these branched surfaces. The idea is to prove that all nontrivial slopes are realized if the knot isn't a cable. A good first step is:

Problem: Prove that for any rational $r\in (-1,1)$ the complement of a nearly-fibered knot admits a CTF foliating the boundary by curves of slope $r$.

The notion of nearly-fibered makes sense (and was defined) for knots in other 3-manifolds as well. Accordingly, another tractable problem related to nearly-fibered knots is:

Problem: Classify genus-1 nearly-fibered knots in other simple manifolds (e.g. the Poincare sphere).



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