arXiv research

The paper finds non-isotopic Legendrian unit conormal bundles in high dimensions.

The paper reformulates Legendrian contact homology using string topology.

Proves Arnol'd's chord conjecture for conormal bundles.

The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^3$ is the submanifold of the cotangent bundle $T^* \mathbb{R}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent bundle $S^* \mathbb{R}^3$, one obtains the unit conormal $Λ_K$, and the Legendrian…

We summarize recent work on a combinatorial knot invariant called knot contact homology. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated to the knot.

The paper finds non-contractible loops of Legendrian tori from knot families.

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift i…

The paper studies knot types of clean intersections in a 3D space.

We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.

A generalization to the almost complex setting of a well-known result by S. Webster is given. Namely, we prove that if $Γ$ is a strongly pseudoconvex hypersurface in an almost complex manifold $(M, J)$, then the conormal bundle of $Γ$ is a totally real submanifold of $(T^*M, \J)$, where $\J$ is the lifted almost comple…

We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through th…

This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special L…

A mathematical isomorphism connects Floer homology to DAHA representations.

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidi…

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves $Γ$. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of $Γ$ covers $T^*M$.

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with cor…

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained.

Study Morse models for torus algebra related to knot homology.

We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.

We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and ce…

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

Heegaard Floer homology connects to polynomial representations of Hecke algebras.

The paper calculates a formula for knot complements using holomorphic curves.

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $Λ_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $Λ_K$, the homology of a di…

Characterizes special curves on surface tangent bundles.

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…

Study shows how certain foliations in unit tangent bundles behave.

We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…

We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.

This article shows that given any orientable 3-manifold X, the 7-manifold T^*X x R admits a closed G_2-structure varphi=Re(Omega)+omega\wedge dt where Omega is a certain complex-valued 3-form on T^*X; next, given any 2-dimensional submanifold S of X, the conormal bundle N^*S of S is a 3-dimensional submanifold of T^*X …

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higher-dimensional tori.

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

Study on Klein bottle's cotangent bundle using contact homology.

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module …

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $η$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $η$-E…

Characterizes magnetic unit vector fields on Lie groups.

The paper proves Morse estimates for translated points on unit tangent bundles.

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

Minimal vector fields on oscillator groups studied, with specific conditions for minimality.

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology a…

Researchers create metrics on hyperbolic space's tangent bundle.

Develops a method to construct entire minimal graphs of odd dimensions.

Paper generalizes Hamiltonian mechanics using line bundles.

Geodesics on modular surface yield arithmetic 3-manifolds.

Unique symplectic fillings of odd spheres' cotangent bundles proven.

The unit ball is characterized by a Kähler-Einstein potential.

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorp…