arXiv subjects
Publications and source records attributed to Viktória Földvári.
We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston--Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.
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 $(M, ξ, \mathcal{F})$ 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 earlier work of Licata and Vértesi. 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.
We prove the following Helly-type result. Let $\mathcal{C}_1,\dots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful selection of $2d$ sets, $C_{i_k}\in \mathcal{C}_{i_k}$ for each $1\leq k\leq 2d$ with $1\leq i_1<\dots<i_{2d}\leq 3d$, the intersection $\bigcap\limits_{k=1}^{2d} C_{i_k}$ is of volume at least 1. Then there is an $1\leq i \leq 3d$ such that $\bigcap\limits_{C\in \mathcal{C}_i} C$ is of volume at least $d^{-O(d^2)}$.
A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $μ$. Here, the homothetic copies of a centrally 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 $\mathbb{R}^d$ 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.
According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant $Υ$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that $Υ$, as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that $Υ$ provides a lower bound on the unknotting number.
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.