# A kind of foliantion on figure eight knot complement

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 transversly with $\partial N$ is $n$ pareller 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)$F$ intersect transversly with $\partial N$ is two pareller 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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You might have a look at the thesis of Timothy Schwider: dl.dropbox.com/u/8592391/dissertation.pdf – Ian Agol Feb 8 '12 at 0:29
@Agol, Thank you. – Bin Yu Feb 8 '12 at 6:53

@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)"). – Bin Yu Feb 7 '12 at 19:52