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Let $N$ be the figure 8 knot complement, What we can say about such kind of dim 2 foliation $F$ on $N$: (1) no Reeb (2 dim); (2) $F$ intersect transversely with $\partial N$ is $n$ parallel Reeb (1 dim)?

Here is a natural example. Let $T^2$ be a torus and $f=(2,1;1,1)$ be a diff on $T^2$. Suppose $(M,\phi_t)$ is the suspension of $(T^2,f)$, which is a transitive Anosov flow. Then cut a standard small solid torus neighborhood of the closed orbit of 0, we obtain a manifold $N$ which is homeomorphic to figure 8 knot complement. The stable manifolds of $\phi_t$ (restrict to $N$) give $N$ a dim 2 foliation $F$ which satisfies: (1) no Reeb; (2) $F$ intersects transversely with $\partial N$ is two parallel Reeb.

This example comes from the paper link text where Franks and Williams constructed the first example of nontransitive Anosov flow.

Can we construct more such kind of foliation? How? Can we classify them in some sense? ...

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Choose a Seifert surface which minimizes the Thurston norm in its relative homology class. (This exists because Seifert surfaces are not 0-homologous.) Gabai's Theorem says that every Thurston-norm minimizing surface is the leaf of a taut foliation. Taut foliations do not have Reeb components. Link

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  • $\begingroup$ @thku, thank you. But, I feel for figure 8 knot, since it is fibered knot, such a foliation intersect with the boundary torus maybe (at least, for the lowest genus seifert surface) pareller circles foliation (doesn't satisfy that "$F$ intersect transversly with $\partial N$ is $n$ pareller Reeb (1 dim)"). $\endgroup$
    – Bin Yu
    Commented Feb 7, 2012 at 19:52

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