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If Y is a closed orientable graph manifold, we show that Y admits a coorientable taut foliation if and only if Y is not an L-space. Combined with previous work of Boyer and Clay, this implies that Y is an L-space if and only if π1(Y) is not left-orderable. [ABSTRACT FROM AUTHOR]

In [2], Cantwell–Conlon introduced a knot invariant, called depth. In this paper, we discuss the depth of a foliation ℱ on a 3-manifold with ℱ being (*) "codimension one, transversely oriented, taut, and proper". We show that for each n > 0, there exists a 3-manifold M such that (minimal depth of ℱ on M with ℱ being (*) and having exactly one depth 0 leaf) is greater than (minimal depth of ℱ on M with ℱ being (*)) + n. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a fibred, compact, orientable 3-manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30. [ABSTRACT FROM PUBLISHER]