This is a companion paper to earlier work of the authors (Preprint, arXiv:1604.03466, 2016), which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We establish a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under (Formula presented.) conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg–Witten theory, Khovanov homology, and (Formula presented.). Many examples are included.
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