Foliations transverse to the turbulized foliations of punctured torus bundles over a circle (Q802175)

The author asks the following question: Does there exist a foliation transverse to the turbulized foliation \(J_ k\) of \(S^ 3\) minus a tubular neighborhood N(k) of a knot k? In [Tôhoku Math. J., II. Ser. 34, 179-238 (1982; Zbl 0513.57015)] the author classified such foliations thereby extending results on foliations transverse to the Reeb foliation [\textit{I. Tamura} and \textit{A. Sato}, Publ. Math., Inst. Hautes Étud. Sci. 54, 5-36 (1981; Zbl 0484.57016)]. The main results (Theorems 1 and 2) are that when k is the trefoil knot, \(J_ k\) doesn't admit a transverse foliation, and when k is the figure eight knot, \(J_ k\) does. These theorems are a result of the author's Theorem 6. Let \(\phi\) be a diffeomorphism of the punctured torus \(T^ 2(1)\); then the associated turbulized foliations (of a punctured torus bundle over \(S^ 1)\) admits a transverse foliation iff the induced homomorphism on \({\mathbb{Z}}\)-homology \(H_ 1(\phi): H_ 1(T^ 2(1))\to H_ 1(T^ 2(1))\) has trace \(\geq 2\).