Essential laminations and Haken normal form

An algorithm to detect laminar 3–manifolds

“…For a given n, there are only finitely many such subdivisions, which may be found combinatorially, because of the linearity of the foliation V restricted to each tetrahedron of τ . For each such subdivision µ, it then enumerates all branched surfaces in µ (2) transverse to V , and checks to see whether they have no sink disk or trivial bubble. The certificate is a description of the subdivision µ and the branched surface in µ (2) .…”

“…For each such subdivision µ, it then enumerates all branched surfaces in µ (2) transverse to V , and checks to see whether they have no sink disk or trivial bubble. The certificate is a description of the subdivision µ and the branched surface in µ (2) . The branched surface B has a cell structure τ .…”

The Gromov norm and foliations

“…Then any minimal genuine lamination Λ (i.e. one with every leaf dense in Λ) can be put in normal form with respect to T , by a theorem of M. Brittenham [3]. For instance, an incompressible surface is an example of a minimal lamination.…”

The simplest branched surfaces for a foliation

“…For example, Gabai [1987] used branched surfaces to construct taut foliations, which allowed him to identify the minimal genus spanning surface for a wide class of knots. Brittenham [1995] showed that a 3-manifold contains an essential lamination if and only if it contains one that is carried (with full support) by one of a finite collection of normal branched surfaces. Agol and Li [2003] used branched surfaces to develop an algorithm for determining if a manifold contains such a lamination.…”