We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, ϕ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T2-bundle over S1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, 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.
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