Slicing planar grid diagrams: a gentle introduction to bordered Heegaard   Floer homology
  We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications. 
Robert Lipshitz, Peter Ozsvath, Dylan Thurston