arXiv research

We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduc…

Torsion found in knot homology, challenging augmentation theories.

The main result of this paper is that, off of a `fundamental class' in degree 1, the linearized Legendrian contact homology obeys a version of Poincare duality between homology groups in degrees k and -k. Not only does the result itself simplify calculations, but its proof also establishes a framework for analyzing coh…

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…

Study properties of contact structures on symplectic disk bundles with concave boundaries.

We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show …

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjec…

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …

If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero …

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally,…

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while pr…

The paper extends knot contact homology to tangles and proves a gluing formula.

The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.

Contact gluing maps are shown to be equivalent in sutured Floer homology.

New contact structures detected by contact homology.

Survey of contact homology for contact manifolds.

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this the…

Survey on proof of homology isomorphism between two complex theories.

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …

The paper examines conditions for contact surgeries on rational homology 3-spheres.

We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.

Motivation. Protein contact map describes the pairwise spatial and functional relationship of residues in a protein and contains key information for protein 3D structure prediction. Although studied extensively, it remains very challenging to predict contact map using only sequence information. Most existing methods pr…

Study examines homology of contact CR-submanifolds in complex Euclidean space.

Cylindrical contact homology computed for links of simple singularities.

Let L be a Legendrian knot in R^3 with the standard contact structure. In [10], a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreove…

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded co…

Persistent Legendrian contact homology distinguishes knots using height functional.

We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.

Study contact invariants using Floer homology to understand knots.

We give a proof of, for the case of contact structures defined by global contact 1-forms, a Theorem stated by Eliashberg that for any overtwisted contact structure on a closed 3-manifold, its contact homology is 0. A different proof is also outlined in the appendix by Yakov Eliashberg.

We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is g…

Let $Λ$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $Λ$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $Λ$, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this she…

In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of…

We consider a family of tight contact structures on the three-dimensional torus and we compute the relative Contact Homology by using the variational theory of critical points at infinity. We will also show some algebraic equivariant homology reductions.

Introduction to contact invariant in bordered Floer homology.

Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higher dimensional contact manifolds and higher order homotopy groups.

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this …

Contact surgeries yield algebraically overtwisted manifolds.

The paper defines contact invariants in bordered Floer homology.

We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó. This version has some clarifications and new figures.

We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology …

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact s…

In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by …

The study finds many tight contact structures on hyperbolic 3-spheres.

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth)…

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, a…

We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure i…

New techniques reveal tight contact manifolds with vanishing contact homology.