Knot Floer homology is an invariant for knots in three-space, defined as a Lagrangian Floer homology in a symmetric product.  It has the form of a bigraded vector space, encoding topological information about the knot.  I will discuss an algebraic approach to computing knot Floer homology, and a corresponding version for links, based on decomposing knot diagrams. This is joint work with Zoltan Szabo, building on earlier joint work (bordered Heegaard Floer homology) with Robert Lipshitz and Dylan Thurston.