Pre-taut sutured manifolds and essential laminations

Abstract

In 1989, D. Gabai and U. Oertel [8] introduced the concept of the essential lamination, which is a hybrid object lying between incompressible surfaces and taut foliations, and generalizing both. We say that a 3-manifold is laminar if it contains an essential lamination. An important result of [8] is that the universal covers of laminar manifolds are homeomorphic to R. This fact furnishes a strong method for studying the manifolds obtained by Dehn surgery along knots, especially concerning Property P Conjecture (nontrivial Dehn surgery on a nontrivial knot in 3 never yields a simply-connected manifold) and Cabling Conjecture (Dehn surgery on a non-cable knot cannot yield a reducible manifold). For example, see [4] for non-torus alternating knots, [3], [12] for 2-bridge knots, [17] for most algebraic knots and [9] for knots with some kind of essential tangle decompositions. We note that by [8] a 3-manifold is laminar if and only if it contains an essential branched surface (for the definition see §2), and the above authors who followed [8] obtained their results by constructing essential branched surfaces. We note that sutured manifold theory was used in [14] and [18]. One of their approaches is to construct a closed essential branched surface in the exterior ( ) of a knot and show that remains essential after any nontrivial Dehn filling along ∂ ( ) (we call such persistently essential). Then we see, by [8], that has Property P in a strong form and that the cabling conjecture is true for . (We say that a knot has strong Property P if every manifold obtained by a nontrivial Dehn surgery along has universal cover R.) It is, however, an open question whether or not every knot with strong Property P admits a persistently essential lamination in its complement. In [1], [2], M. Brittenham had a paradigm shift in proving strong Property P for knots. Instead of constructing a branched surface in the complement of a given knot, he first constructed a branched surface and then embedded a knot in its complement. More precisely, he first constructed a closed branched surface in 3 from any in-