Solves a specific question in mapping class group theory.
problem A specific question in mapping class group theory.
method Analyzes a paper by Etnyre and Van Horn-Morris.
result Answers a specific question in mapping class group theory.
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32 results for “Etnyre”
The paper derives a signature formula for Lefschetz fibrations with planar fibers.
problem Proving a signature formula for Lefschetz fibrations with planar fibers.
method Using theorems of Eliashberg and McDuff, computing Maslov index, and applying Wall's non-additivity formula.
result Derives a signature formula for allowable Lefschetz fibrations over with planar fiber.
In this note we observe that the no two of the three invariants defined for contact structures by Etnyre and Ozbagci -- that is, the support genus, binding number and support norm -- determine the third.
Recently, there have been several breakthroughs in the classification of tight contact structures. We give an outline on how to exploit methods developed by Ko Honda and John Etnyre to obtain classification results for specific examples of small Seifert manifolds.
Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre.
We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .
We prove that every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact surgery of coefficient plus or minus 1 along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in slightly stronger form).
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless)…
This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to filli…
Affirmative answer to a question about symmetric mapping classes and positive factorizations.
problem Whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid.
method Investigated a specific question from a previous paper, answered affirmatively under certain conditions.
result Affirmative answer to the question for mapping classes satisfying certain cyclic conditions.
Study shows 3D shapes can't fit in certain 4D spaces.
problem Proving 3-manifolds can't embed in symplectic 4-manifolds.
method Constructing L-spaces and using Heegaard Floer correction terms, instanton moduli spaces.
result Infinitely many 3-manifolds cannot embed in symplectic 4-manifolds.
In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pin points the error. In "An addendum on iterated torus knots" a more subtle result is proven resulting in giving a geometric realization of the Honda-Etny…
Study contact invariants using Floer homology to understand knots.
problem Understanding contact invariants of knots through surgeries.
method Used Heegaard Floer based contact invariants, sutured Floer homology, and limit constructions.
result Established relations between contact invariants of knots through surgeries.
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 …
Concordance maps in math.GT
A new map from knot concordance to Heegaard Floer homology.
problem Understanding how knot concordance affects Heegaard Floer homology.
method Using sutured Floer homology, we define a module homomorphism that preserves gradings.
result The constructed map respects Alexander and Maslov gradings.
We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specif…
Higher-dimensional contact manifolds lack certain properties.
problem Characterizing properties of higher-dimensional contact manifolds.
method Symplectic handle attachment and alternative proof techniques.
result They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds.
The abstract introduces a new duality via LSFT algebra.
problem Legendrian knot duality and its extension.
method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of bimodules over .
result Explicit construction of homotopy inverse for the Sabloff map.
New contact structures on spheres and tori with weakly compatible metrics.
problem Contact sphere theorem does not hold for weakly compatible metrics.
method Construction of nonvanishing curl eigenfields using Jacobi or trigonometric polynomials.
result Geometric rigidity for tight contact structures on the 3-sphere.
New invariant shows non-zero transverse links in open books.
problem Transverse invariants in open book structures.
method Knot Floer homology and transverse link invariants.
result Transverse link invariant is always nonzero.
The study distinguishes exotic R^4's using knot Floer homology and Casson handles.
problem Distinguishing exotic R^4's with different knot Floer homologies.
method Using Gadgil's end Floer homology and Casson handles.
result A countably infinite family of pairwise nondiffeomorphic chiral exotic R^4's.
A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse s…
The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) a…
We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded …
In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…
In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is false, and the Erratum math.GT/0610565 identifies the gap. The purpose of this paper…
Quantifies tightness in 3D contact manifolds using sub-Riemannian metrics.
problem Estimating maximal tight neighbourhoods of Reeb orbits on 3D contact manifolds.
method Sub-Riemannian metrics and geometric constructions.
result Sharp estimates of tightness radius in terms of Schwarzian derivative bounds.
Study of panhandle polynomials of torus links with geometric applications.
problem Characterizing the HOMFLY-PT polynomial of torus knots and links.
method Utilizing quantum group representations and the Rosso-Jones formula.
result Established panhandle-like structure of HOMFLY-PT polynomials for torus knots and links.
Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in , 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 embeddings of surfaces in 4-manifolds and their mapping classes.
problem Characterizing and understanding embeddings of surfaces in 4-manifolds and their mapping classes.
method Analyzing smooth proper embeddings and mapping classes induced by diffeomorphisms of 4-manifolds.
result Most surfaces do not admit flexible embeddings in 4-manifolds with specific homology types.
Contact structures are induced by nondegenerate skew fibrations of R^3.
problem Understanding contact structures induced by skew fibrations of R^3.
method Characterizing nondegenerate fibrations and using a recent result on tightness in contact metric 3-manifolds.
result The plane field induced by any nondegenerate fibration is a tight contact structure.
New algebraic construction of knot contact homology using perverse sheaves.
problem Algebraic construction of knot contact homology.
method Defining a DG category with a braid group action on perverse sheaves.
result The endomorphism algebra of a distinguished object in the category matches fully noncommutative knot DGA.