We introduce the notion of a nested open book, a submanifold equipped with an open book structure compatible with an ambient open book, and describe in detail the special case of a push-off of the binding of an open book. This enables us to explicitly describe a natural open book decomposition of a fibre connected sum …
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48 results for “open books”
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
The abstract discusses embedding manifolds in open books and contact structures.
problem Embedding manifolds in open books and contact structures.
method Using open book embeddings and contact structures, the abstract proves embedding theorems for various manifolds.
result Closed manifolds can be embedded in open books and contact structures.
Coverings of open booksmath.GT
We study a coverings of open books and virtually overtwisted contact manifolds using open book foliations. We show that open book coverings produces interesting examples such as transverse knots with depth grater than 1. We also demonstrate explicit examples of virtually overtwisted open books.
An open book decomposition of a 3-manifold induces a Heegaard splitting for , and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of . It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of …
In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact struc…
New method shows links can be braided open book bindings.
problem Understanding fibered links and their bindings.
method Mutual arc presentations and braided open books.
result Every fibered link is the binding of a braided open book.
New examples show high twisting doesn't guarantee open book maximality.
problem Maximality of open book pages under high twisting.
method Double branched covers of closed braids.
result Arbitrarily high twisting does not guarantee maximal Euler characteristic.
We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
Foliated Open Booksmath.GT
This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation on its boundary. These \textit{foliated open books} offer a finer tool for studying contact manifolds with convex boundary than existing models, as the boundary foliation ca…
In this note, we discuss embeddings of --manifolds via open books. First we show that every open book of every closed orientable --manifold admits an open book embedding in any open book decompistion of and with the page a disk bundle over and monodromy the iden…
Simplicial volume vanishes for 4-manifolds with open book decompositions.
problem Characterizing open book decompositions in dimension 4.
method Adapted proof by Bucher and Neofytidis.
result Simplicial volume vanishes for 4-manifolds with open book decompositions.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
Common positive stabilisation found for isotopic contact structures.
problem Isotopic contact structures and open book decompositions.
method Sequence of positive stabilisations and destabilisations.
result Common positive stabilisation exists for isotopic contact structures.
New proof of Giroux Correspondence for tight contact 3-manifolds.
problem Proving the Giroux Correspondence for tight contact 3-manifolds.
method Introducing tight Heegaard splittings, using refinement process, and translating moves between splittings to moves between open books.
result Proves the tight Giroux Correspondence for contact 3-manifolds.
The authors introduce Morse foliated open books for studying contact manifolds.
problem Studying contact manifolds with boundary.
method Introducing Morse foliated open books and extending the right-veering concept.
result Right-veering plays a similar role in detecting overtwistedness in foliated open books.
As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.
Proves codimension 2 spun embedding for specific manifolds.
problem Codimension 2 spun embedding for manifolds.
method Proves embedding using open book decompositions and spin structures.
result Proves spun embedding for simply connected spin 5-manifolds and 3-dimensional open books.
New 4D shapes can't be opened like books.
problem Existence of 4D shapes without open book decompositions.
method Demonstrated existence of infinitely many parallelizable 4-manifolds without open book decompositions.
result No open book decomposition for certain Engel manifolds.
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…
Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.
The study explores nonorientable 3-manifolds using open books and their monodromies.
problem Investigating open books for nonorientable 3-manifolds.
method Analyzing monodromies of open books for specific nonorientable 3-manifolds.
result Infinitely many nonisotopic genus two open books for and .
New open books defy positive factorisation in genus one.
problem Understanding Stein fillable contact structures and their monodromies.
method Constructing an infinite family of open books with specific properties.
result Monodromies of constructed open books do not admit positive factorisations.
In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components…
New 'book links' generalize braids and plats, proving Markov's theorem.
problem Generalizing braid theory to new structures.
method Extending open book foliations to prove Markov's theorem for new objects.
result Proved Markov's theorem in a broader context of 'book links'.
New tools classify symplectic fillings of contact 3-manifolds.
problem Classifying symplectic fillings of contact 3-manifolds.
method Spinal open book decompositions and bordered Lefschetz fibrations.
result Symplectic fillings of contact 3-manifolds are deformation equivalent to complements of positive multisections in bordered Lefschetz fibrations.
We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation. As applications, we study how open book foliations change under a stabilization of the op…
Algorithm trisects 4D book into three chapters.
problem Trisecting 4D book into chapters.
method Algorithm for open book decompositions of 4-manifolds.
result Algorithm works for various types of manifolds.
We show that if the monodromy of an open book decomposition has sufficiently high displacement distance, acting on the loop and arc complex for a page, then it is the unique minimal Euler characteristic open book for the manifold. In particular, we show that such an open book induces the unique (up to isotopy) minimal …
We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called ``twisting loop''. We show that the existence of a twisting loop on the fiber surface of an open book is e…
Study four ways to define braided open book decompositions of the 3-sphere.
problem Define and compare different ways to make open book decompositions of the 3-sphere 'braided'.
method Analyze four specific definitions of braided open book decompositions and prove their equivalence.
result All four definitions of braided open book decompositions are equivalent.
Detects right-veering properties in open books using combinatorial methods.
problem Determining the right-veering property of contact structures.
method Combinatorial approach to detect left-veering arcs in open books.
result Existence of an algorithm to detect right-veering for compact surfaces.
Open books constructed from Morse functions and divides are shown to be isotopic.
problem Isotoping open books constructed from Morse functions and divides.
method Showed isotopy of open books constructed from Morse functions and divides.
result Open books are isotopic when constructed from Morse functions and divides.
In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this re…
Given two open books with equal pages we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monod…
Open books and plumbingsmath.GT
We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book for with relatively low genus. Thus we produce open books with low genus p…
We exhibit infinitely many overtwisted, right-veering, non-destabilizable open books, thus providing infinitely many counterexamples to a conjecture of Honda-Kazez-Matic. The page of all our open books is a four-holed sphere and the underlying 3-manifolds are lens spaces.
If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. We define a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and we discuss various applications. In particular, we determine the su…
We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is necessary up to stabilization by positive Hopf-bands. Using these techniques we prove …
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.
We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.
In the present article we determine and characterize completely the support genus, the binding number and the norm of a page of an open book under the following restrictions: M is a rational homology sphere which can be realized as the link of a surface singularity. Moreover, we restrict ourselves to the collection of …
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…
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 describe explicit horizontal open books on some Seifert fibered 3--manifolds. We show that the contact structures compatible with these horizontal open books are Stein fillable and horizontal as well. Moreover we draw surgery diagrams for some of these contact structures.
Compatible Relative Open Books on Relative Contact Pairs via Generalized Square Bridge Diagramsmath.GT
Article generalizes open book construction for 5D contact pairs.
problem Constructing compatible open books on relative contact pairs.
method Introduces generalized square bridge position for 5D Legendrian links.
result Algorithm constructs relative open book decompositions on relative contact pairs.
Homogeneous braids are visually prime, solving a Cromwell question.
problem Addressing Cromwell's question about visually prime braids.
method Using a criterion for primeness of open books and Murasugi sums.
result Closures of homogeneous braids are visually prime.