Manifolds with small Heegaard Floer ranks

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil.We classify links whose branched double cover gives rise to this manifold.Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.