Link Floer homology detects the Thurston norm

We prove that, for a link L in a rational homology 3–sphere, the link Floer homology detects the Thurston norm of its complement. This result has been proved by Ozsváth and Szabó for links in S^3. As an ingredient of the proof, we show that knot Floer homology detects the genus of null-homologous links in rational homology spheres, which is a generalization of an earlier result of the author. Our argument uses the techniques due to Ozsváth and Szabó, Hedden and the author