  Knot Floer homology is a knot invariant defined using holomorphic curves. In
more recent work, taking cues from bordered Floer homology,the authors
described another knot invariant, called "bordered knot Floer homology", which
has an explicit algebraic and combinatorial construction. In the present paper,
we extend the holomorphic theory to bordered Heegaard diagrams for partial knot
projections, and establish a pairing result for gluing such diagrams, in the
spirit of the pairing theorem of bordered Floer homology. After making some
model calculations, we obtain an identification of a variant of knot Floer
homology with its algebraically defined relative. These results give a fast
algorithm for computing knot Floer homology.
