| In the first part of this thesis, we study the contact structures compatible with open books (T, &phis;) where T has genus one and one boundary component. Every diffeomorphism, &phis; , of T which restricts to the identity on ∂ T is isotopic to a product of Dehn twists along dual, non-separating curves, x and y, in S. Given such a product, we supply an algorithm which determines whether the contact structure compatible with the open book (T, &phis;) is tight or overtwisted, for all but a small family of reducible diffeomorphisms &phis;.; In addition, we explore the Heegaard Floer homology of 3-manifolds with open book decompositions (T, &phis;). Let us denote such a manifold by MT,&phis;. These are precisely those 3-manifolds which contain genus one fibered knots, or, equivalently, those 3-manifolds which arise as branched double covers of S3 along closed 3-braids. Given a product, &phis;, of Dehn twists along x and y, we supply an algorithm which computes the Heegaard Floer homology of MT,&phis; whenever MT,&phis; is a rational homology 3-sphere. Along the way, we compute the Heegaard Floer homology of every T 2-bundle over S1 with first Betti number equal to one, and we show that our computations agree with Lebow's computations of embedded contact homology for all such torus bundles with pseudo-Anosov monodromy. We use these results to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.; Finally, we uncover a novel naturality feature of the Ozsvdath-Szabo contact invariant. For a compact surface with boundary, S, we show that there is a comultiplication map D&d5;:HF&d14; &parl0;-MS,hg&parr0;→HF&d14; &parl0;-MS,g&parr0;⊗HF&d14; &parl0;-MS,h&parr0; which sends the contact invariant c(S, hg ) to c(S, g) ⊗ c(S, h). It follows that if the contact structures compatible with the open books (S, g) and (S, h) are strongly-fillable then the contact structure compatible with the open book (S, hg) is tight. We extend this comultiplication to a map on H F+ and, as a result, we obtain information about the support genus of the contact structure compatible with (S, hg) in terms of the open books (S, g) and (S, h). |
