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Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $$(M, \xi , \mathcal {F})$$ whose convex boundary is equipped with a signed singular foliation $$\mathcal {F}$$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $$c_D$$ and $$c_A$$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $$\mathcal {F}$$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić. 

Given a knot K in S3, let u−(K) (respectively, u+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l−(K),l+(K) and l(K), which give lower bounds on u−(K),u+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and satisfies l(K) ≥ ν+(K), while the difference l(K) − ν+(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.