TAUT FOLIATIONS IN PUNCTURED SURFACE BUNDLES, II

Abstract

Given a fibered 3-manifold M, we investigate exactly which boundary slopes can be realized by perturbing fibrations along product discs. Since these perturbed fibrations cap off to give taut foliations in the corresponding surgery manifolds, we obtain surgery information. For example, recall that a knot k is said to satisfy Property P if no finite surgery along k yields a simply-connected 3-manifold. We show that if a non-trivial fibered knot $k\subset S^3$ fails to satisfy Property P, then necessarily k is hyperbolic with degeneracy slope $\pm\frac{2}{1}$. When k is hyperbolic and $d(k)=\frac{2}{1}$ (respectively, $-\frac{2}{1}$), we show that the only candidate for a counterexample to Property P is surgery coefficient $\frac{1}{1}$ (respectively, $-\frac{1}{1}$).

2000 Mathematical Subject Classification: primary 57M25; secondary 57R30.