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, ξ, F) whose convex boundary is equipped with a signed singular foliation 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 cD and cA 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 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. CONTENTS1. Introduction 1 2. Preliminaries 4 3. Construction of the contact invariant 18 4. Invariance 23 5. Gluing 29 6. Relationship to the HKM contact element 33 References 36 2010 Mathematics Subject Classification. 57M27.

No abstract

The $$\mu $$ μ -kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of $$\mu $$ μ . As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order $$\mu $$ μ . Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the $$\mu $$ μ -kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For d-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is $$3^d$$ 3 d . The case $$\mu =1$$ μ = 1 is the Bezdek–Pach Conjecture, which asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $$\mathbb R^d$$ R d is $$2^d$$ 2 d . We verify the conjecture on the plane, that is, when $$d=2$$ d = 2 . Indeed, we show that the number in question is four for any planar convex body.

No abstract

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant [Formula: see text] without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that [Formula: see text], as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that [Formula: see text] provides a lower bound on the unknotting number.

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.

By examining knot Floer homology, we extend a result of Ozsváth and Stipsicz and show further infinitely many Legendrian and transversely non-simple knot types among two-bridge knots. We give sufficient conditions of Legendrian and transverse non-simplicity on the continued fraction expansion of the corresponding rational number.

The µ-kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of µ. As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order µ. Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the µ-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For d-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is 3 d . For µ = 1 the general version yields to another known problem: The Bezdek-Pach Conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in R d is 2 d . We verify the conjecture on the plane, that is, when d = 2. Indeed, we show that the number in question is four for any planar convex body.