arXiv research

New Alexander polynomial defined for transverse graphs.

We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same tr…

Study of Poincaré-Reeb graphs for algebraic domains.

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …

A graph helps understand Artin groups better.

In this paper, we introduce two notions on a surface in a contact manifold. The first one is called degree of transversality (DOT) which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a compar…

Grid homology properties for MOY graphs studied.

Invariant obstructs separating coassociative 4-folds.

Study on planar graphs in Poincare model of hyperbolic geometry.

Study connects flow dynamics to 3D geometry via surface intersections.

New polynomial helps compute flow growth rates in 3D manifolds.

Riemannian manifolds can be realized as leaf spaces of matchbox manifolds.

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…

In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we give a theorem of Viro's patchworking type to build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. We use this result to build a family …

A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let …

We prove that any complete surface with constant mean curvature in a homogeneous space E(κ,τ) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in E(κ,τ) (different fr…

CMS uses machine learning to improve particle flow reconstruction.

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…

I answer an open question left by Gui-Song Li in "On self-intersections of immersed surfaces" (AMS Proceedings, Volume 126, 1998, pp.3721-3726.) The intersection graph $M(i)$ of a generic surface $i:F \to S^3$ is the set of values which are either singularities or intersections. It is a multigraph whose edges are trans…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at…

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph $G = G(F)$ of a pseudo-Riemannian foliation there exis…

Study on transverse Ricci solitons on compact foliated manifolds.

New examples show transverse knots are determined by their branched covers.

Defines transverse symbols for foliated manifolds and proves their K-homology class.

Study transverse Dolbeault cohomology for almost complex structures.

We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse…

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…

Study finds non-isotopic transverse tori in Engel manifolds.

New theorem allows transverse links to be braided with rational book structure.

Reidemeister's theorem proved using smooth functions and transversality.

The study classifies transverse spheres in flag manifolds and finds new examples.

We study the transverse Lusternik-Schnirelmann category of a Riemannian foliation on a compact manifold. We obtain a necessary and sufficient condition when the transverse LS category is finite. We also introduce a variation on the concept of transverse LS category, the essential transverse category, and show that this…

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.

In 2003, Ozsváth and Szabó defined the concordance invariant $τ$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $τ$ for knots in $S^3$ and a combinatorial proof that $τ$ gives a lower bound for the slice genus of a knot. Recently, Har…

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…

Study on geometry and dynamics of transverse subgroups.

Engel manifolds show transverse tori can be made to have various formal invariants.

We classify positive transversal torus knots in tight contact structures up to transversal isotopy.

We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.

A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are isotopic as transverse knots are also isotopic as transverse braids. The methods of the…

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler cha…

A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G a 3-valent graph in M was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected s…

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide …

In the present paper, we introduce screen transversal lightlike submanifolds of metallic semi-Riemannian manifolds with its subclasses, namely screen transversal anti-invariant, radical screen transversal and isotropic screen transversal lightlike submanifolds, and give an example. We show that there do not exist co-is…