Taut foliations in knot complements

A Higher Dimensional Generalization of Taut Foliations

A higher dimensional generalization of taut foliations is introduced. Tools from symplectic geometry are used to describe surgery constructions, and to study the space of leaves of this class of foliations. Codimension one foliations are too large a class of structures to obtain strong structure theorems for them. According to a theorem of Thurston [39] a closed manifold admits a codimension on...

Geodesic Surfaces in Knot Complements

Branching in the Universal Cover of Taut Foliations

We study the topology of codimension one taut foliations of closed orientable 3-manifolds which are smooth along the leaves. In particular, we focus on the lifts of these foliations to the universal cover, specifically when any set of leaves corresponding to nonseparable points in the leaf space can be totally ordered. We use the structure of branching in the lifted foliation to find conditions...

Graph Manifolds Z-homology 3-spheres and Taut Foliations

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either S or Σ(2, 3, 5) admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-oriented taut foliation, are equivalent for graph manifold Z-homology 3-spheres.

Floer Homology and Knot Complements

We use the Ozsváth-Szabó theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the Ozsváth-Szabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on ...