Study shows infinitely many non-simple Legendrian knots in two-bridge knots.
problem Characterizing non-simple Legendrian knots in two-bridge knots.
method Using knot Floer homology and continued fraction expansions.
result Infinitely many Legendrian and transversely non-simple knot types among two-bridge knots.
We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.
The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers …
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formula…
Study on transverse Ricci solitons on compact foliated manifolds.
problem Characterizing transverse Ricci solitons on compact foliated manifolds.
method Investigation of self-similar solutions of the transverse Ricci flow, analysis of taut Riemannian foliations.
result Established relations between taut Riemannian foliations and transverse Ricci solitons, found examples of transverse Ricci solitons.
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
Defines transverse symbols for foliated manifolds and proves their K-homology class.
problem Transverse index theory for foliated manifolds.
method Using filtrations of tangent bundles, defining transverse symbols, and constructing equivariant KK-classes.
result Transversally Rockland operators yield a K-homology class and there is a Poincare duality result.
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.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
Paper characterizes generic transversality, improving on Mather's result.
problem Understanding and defining generic transversality.
method Characterization of transversality based on Mather's work.
result Improves on Mather's transversality result.
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
The study classifies transverse spheres in flag manifolds and finds new examples.
problem Classifying transverse spheres in flag manifolds.
method Using topological K-theory and constructions of transverse spheres.
result Classification of transverse spheres in various flag manifolds.
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 …
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.
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 R3, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
Study on geometry and dynamics of transverse subgroups.
problem Understanding the geometry and dynamics of transverse subgroups.
method Survey of recent research on semi-simple Lie groups.
result Recent findings on transverse subgroups of semi-simple Lie groups.
Study on transverse invariant from Khovanov homology and its properties.
problem Understanding the relationship between knot isotopy and transverse isotopy.
method Analyzing fractional Dehn twist coefficients, quasipositivity, and stability of transverse invariants.
result Stability and triviality of the transverse invariant for specific families of braids and knots.
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 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.
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 …
Study on lightlike submanifolds in metallic semi-Riemannian manifolds.
problem Characterizing and investigating properties of lightlike submanifolds in metallic semi-Riemannian manifolds.
method Introduced and analyzed subclasses of screen transversal lightlike submanifolds and investigated their geometric properties.
result Necessary and sufficient condition for an isotropic screen transversal lightlike submanifold to be totally geodesic.
New method computes transverse knot invariant using braids.
problem Compute transverse invariant of knots and links.
method Define new combinatorial complex to compute knot Floer homology.
result Identify BRAID invariant of transverse knots and links.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field
Study transverse and Legendrian invariants of certain link cables using Floer homology.
problem Transverse and Legendrian invariants of specific link cables.
method Combining grid diagrams and Floer homology, using inclusion maps of grid complexes.
result Transverse and Legendrian invariants of certain link cables are zero for large q/p. Study classifies twist knots with maximal self-linking number in S^3.
problem Classifying transverse-universal knots in S^3.
method Classification of transverse twist knots with maximal self-linking number.
result Obtained an infinite family of non-transverse-universal knots.
Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
New conditions found for Riemannian foliations with specific conformal fields.
problem Conditions for Riemannian foliations with transversal conformal fields.
method Analyzes conditions for transversal isometry of foliations.
result Found conditions for F to be isometric to the sphere. Study links in contact manifolds using open books and overtwisted disks.
problem Understanding transverse links in contact manifolds.
method Using open book decompositions and overtwisted disks, the support genus of transverse links is studied.
result The support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it.
Study of cosmetic contact surgeries on knots, excluding some cases.
problem Determining cosmetic contact surgeries on transverse knots.
method Analyzing contact surgeries and their outcomes on transverse knots.
result Exclude non-trivial cosmetic contact surgeries for certain transverse knots.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.
problem Transversality and regularity of minimal submanifolds and area-minimizing currents.
method Proves transversality and regularity for generic metrics using Baire category and minimization properties.
result Generic metrics ensure transversality and regularity for minimal submanifolds and area-minimizing currents.
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.
Study shows boundary measurements can determine transversal singularities in anisotropic geometries.
problem Determining transversal singularities in anisotropic geometries from boundary measurements.
method Geometric condition on transversal manifold and FBI type transform.
result Recovery of transversal singularities in the linearized problem.
Two new transversality theorems for linearly perturbed mappings are proven.
problem Understanding the behavior of mappings under small perturbations.
method Proves two transversality theorems for generic linearly perturbed Cr mappings. result Establishes new theorems for the stability of mappings under small changes.
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…