This book chapter is published open access.
Manifolds with torus boundary have played a special role in the study of Floer homology for 3-manifolds since the early days of the subject. In joint work with Jonathan Hanselman and Liam Watson, we defined a geometrical Heegaard Floer invariant for 3-manifolds with torus boundary. The invariant is a reformulation of the bordered Floer homology of Lipshitz, Ozsváth, and Thurston, and takes the form of a collection of immersed closed curves (possibly decorated with local systems) in a covering space of the punctured torus. We briefly discuss the construction of the invariant and some applications to the L-space conjecture of Boyer–Gordon–Watson and Juhász. We then describe a generalization to manifolds with sutured boundary, and some applications to the study of satellite knots.