Floer homology and surface decompositions

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, \gamma)--> (M', \gamma') is a sutured manifold decomposition then SFH(M',\gamma') is a direct summand of SFH(M, \gamma). To prove the decomposition formula we give an algorithm that computes SFH(M,\gamma) from a balanced diagram defining (M,\gamma) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, \gamma) is taut then SFH(M,\gamma) is non-zero. Other applications include simple proofs of a result of Ozsvath and Szabo that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry. Moreover, using these methods we show that if K is a genus g knot in a rational homology 3-sphere Y whose Alexander polynomial has leading coefficient a_g non-zero and if the rank of \hat{HFK}(Y,K,g) < 4 then the knot complement admits a depth < 2 taut foliation transversal to the boundary of N(K).Comment: 40 pages, 10 figures. Improved, expanded expositio

Laminar Branched Surfaces in 3-manifolds

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are standard in the theory of branched surfaces, and a sink disk is a disk branch of the branched surface with all branch directions of its boundary arcs pointing inwards. We will show in this paper that every laminar branched surface carries an essential lamination, and any essential lamination that is not a lamination by planes is carried by a laminar branched surface. This implies that a 3-manifold contains an essential lamination if and only if it contains a laminar branched surface.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper6.abs.htm

The sutured Floer homology polytope

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.Comment: 37 pages, 4 figures, improved expositio

Link Floer homology detects the Thurston norm

We prove that, for a link $L$ in a rational homology 3--sphere, the link Floer homology detects the Thurston norm of its complement. This generalizes the previous results due to Ozsv\'ath, Szab\'o and the author.Comment: 25 pages, 1 figur

Holomorphic discs and sutured manifolds

In this paper we construct a Floer-homology invariant for a natural and wide class of sutured manifolds that we call balanced. This generalizes the Heegaard Floer hat theory of closed three-manifolds and links. Our invariant is unchanged under product decompositions and is zero for nontaut sutured manifolds. As an application, an invariant of Seifert surfaces is given and is computed in a few interesting cases.Comment: This is the version published by Algebraic & Geometric Topology on 4 October 200

Sutured Heegaard diagrams for knots

We define sutured Heegaard diagrams for null-homologous knots in 3-manifolds. These diagrams are useful for computing the knot Floer homology at the top filtration level. As an application, we give a formula for the knot Floer homology of a Murasugi sum. Our result echoes Gabai's earlier works. We also show that for so-called 'semifibred' satellite knots, the top filtration term of the knot Floer homology is isomorphic to the counterpart of the companion.Comment: This is the version published by Algebraic & Geometric Topology on 2 April 200

An algorithm to detect laminar 3-manifolds

We show that there are algorithms to determine if a 3-manifold contains an essential lamination or a Reebless foliation.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper8.abs.htm

Bounds on exceptional Dehn filling

We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper15.abs.htm

Taut ideal triangulations of 3-manifolds

A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behaviour is very similar to that of a taut foliation. For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai's result that the singular genus of a knot in the 3-sphere is equal to its genus.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper12.abs.htm