Constructing taut foliations
In [Contemp. Math. 20, 131-143 (1983; Zbl 0524.57004), 44, 93-100 (1985; Zbl 0584.57003); J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013), 26, 461-478 (1987; Zbl 0627.57012), 479-536 (1987; Zbl 0639.57008); Topology 23, 381-394 (1984; Zbl 0567.57021), 26, 209-210 (1987; Zbl 0621.57004); Comment. Math. Helv. 61, 519-555 (1986; Zbl 0621.57003); Genera of the arborescent links, Mem. Am. Math. Soc. 339, 1-98 (1986; Zbl 0585.57003)] \textit{David Gabai} uses taut foliations to prove an impressive collection of results. In this paper, we use a refinement of Gabai's techniques to construct taut foliations which permit a pleasingly simple description. For certain knots \(k\), the foliations constructed allow one to conclude that \(k\) has Property P: no nontrivial Dehn surgery on \(k\) yields a simply-connected manifold. In particular, it follows that a large class of alternating knots satisfies Property P. More precisely, we obtain the following Theorem 0.1. Let \(k\) be a knot in \(S^3\) with alternating projection in which there is nontrivial nesting among the Seifert cycles. Then any manifold obtained from \(S^3\) by nontrivial Dehn surgery along \(k\) contains a taut foliation. Hence, nontrivial Dehn surgery along \(k\) yields a manifold with infinite fundamental group. Theorem 0.2. Let \(k\) be a nontrivial knot in \(S^3\) with alternating projection in which there is no nesting among the Seifert cycles. Then either all half-twists are positive and any manifold obtained from \(S^3\) by Dehn surgery along \(k\) with nonnegative surgery coefficient contains a taut foliation or else all half-twists are negative and any manifold obtained by Dehn surgery with nonpositive surgery coefficient contains a taut foliation.
scientific article; zbMATH DE number 852027
scientific article