We use bordered Floer homology to study 3-manifolds obtained by gluing together two knot complements, in view of several conjectures concerning the classification of L-spaces, manifolds with the simplest possible Heegaard Floer homology. We show that the homology sphere obtained by splicing the complements of nontrivial knots in the three-sphere must have Floer homology of rank greater than one. By extending this approach to knots in arbitrary homology spheres, we hope to prove that a manifold whose Heegaard Floer homology has rank one cannot contain an incompressible torus. Using related techniques, we give examples of manifolds that are not L-spaces but which do not admit taut foliations. This is joint work with Matthew Hedden.