scholarly journals Canonical contact unit cotangent bundle

2018 ◽  
Vol 18 (4) ◽  
pp. 405-424
Author(s):  
Takahiro Oba ◽  
Burak Ozbagci
Keyword(s):  
Cotangent Bundle ◽  
Contact Structure ◽  
Closed Surface ◽  
Open Book ◽  
Open Book Decomposition ◽  
Contact Unit

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

scholarly journals On the contact Ozsváth-Szabó invariant

2010 ◽  
Vol 47 (1) ◽  
pp. 90-107
Author(s):  
Tolga Etgü ◽  
Burak Ozbagci
Keyword(s):  
Homology Group ◽  
Contact Structure ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Open Book ◽  
Open Book Decomposition ◽  
Heegaard Diagram ◽  
Contact Surgery ◽  
Heegaard Floer

Sarkar and Wang proved that the hat version of Heegaard Floer homology group of a closed oriented 3-manifold is combinatorial starting from an arbitrary nice Heegaard diagram and in fact every closed oriented 3-manifold admits such a Heegaard diagram. Plamenevskaya showed that the contact Ozsváth-Szabó invariant is combinatorial once we are given an open book decomposition compatible with a contact structure. The idea is to combine the algorithm of Sarkar and Wang with the recent description of the contact Ozsváth-Szabó invariant due to Honda, Kazez and Matić. Here we observe that the hat version of the Heegaard Floer homology group and the contact Ozsváth-Szabó invariant in this group can be combinatorially calculated starting from a contact surgery diagram. We give detailed examples pointing out to some shortcuts in the computations.

scholarly journals Flat plumbing basket and contact structure

2021 ◽  
Vol 30 (02) ◽  
pp. 2150010
Author(s):  
Tetsuya Ito ◽  
Keiji Tagami
Keyword(s):  
Lower Bounds ◽  
Contact Structure ◽  
Seifert Surface ◽  
Open Book ◽  
Open Book Decomposition ◽  
Knots And Links ◽  
Torus Links

A flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3-sphere. In this paper, we examine close connections between flat plumbing baskets and the contact structure supported by the open book. As an application we give lower bounds for the flat plumbing basket numbers and determine the flat plumbing basket numbers for various knots and links, including the torus links.

A NOTE ON OPEN BOOK EMBEDDINGS OF -MANIFOLDS IN

2021 ◽  
pp. 1-8
Author(s):  
SUHAS PANDIT ◽  
SELVAKUMAR A
Keyword(s):  
Open Book ◽  
Open Book Decomposition ◽  
Book Embeddings

Abstract In this note, we show that given a closed connected oriented $3$ -manifold M, there exists a knot K in M such that the manifold $M'$ obtained from M by performing an integer surgery admits an open book decomposition which embeds into the trivial open book of the $5$ -sphere $S^5.$

COMPLEXITY OF OPEN BOOK DECOMPOSITIONS VIA ARC COMPLEX

2010 ◽  
Vol 19 (01) ◽  
pp. 55-69 ◽  
Author(s):  
TOSHIO SAITO ◽  
RYOSUKE YAMAMOTO
Keyword(s):  
Fiber Surface ◽  
Heegaard Splitting ◽  
Open Book ◽  
Open Book Decomposition ◽  
Open Book Decompositions

Based on Hempel's distance of a Heegaard splitting, we define a certain kind of complexity of an open book decomposition, called a translation distance, by using the arc complex of its fiber surface. We then show that an open book decomposition is of translation distance at most two if it is split into "simpler" open book decompositions, or at most three if it admits a Stallings twist on it.

scholarly journals EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS

2006 ◽  
Vol 17 (09) ◽  
pp. 1013-1031 ◽  
Author(s):  
TOLGA ETGÜ ◽  
BURAK OZBAGCI
Keyword(s):  
Contact Structure ◽  
Complex Surface ◽  
Contact Structures ◽  
Open Book ◽  
Open Books ◽  
Tight Contact ◽  
Weinstein Conjecture ◽  
Circle Bundles ◽  
Surface Singularities

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, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

scholarly journals The Casson–Walker invariant of 3-manifolds with genus one open book decompositions

2019 ◽  
Vol 28 (06) ◽  
pp. 1950018
Author(s):  
Atsushi Mochizuki
Keyword(s):  
Mapping Class Group ◽  
Central Extension ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Open Book ◽  
Open Book Decomposition ◽  
Open Book Decompositions ◽  
Framed Link ◽  
Genus One

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

scholarly journals Trisections of 4-manifolds with boundary

2018 ◽  
Vol 115 (43) ◽  
pp. 10861-10868 ◽  
Author(s):  
Nickolas A. Castro ◽  
David T. Gay ◽  
Juanita Pinzón-Caicedo
Keyword(s):  
Existence Result ◽  
Manifolds With Boundary ◽  
Open Book ◽  
Original Proof ◽  
Open Book Decomposition ◽  
Manifold With Boundary ◽  
The Given

Given a handle decomposition of a 4-manifold with boundary and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components and show how trisected 4-manifolds with multiple boundary components glue together.

scholarly journals A note on overtwisted contact structures

2011 ◽  
Vol 48 (1) ◽  
pp. 130-134
Author(s):  
El.if Yilmaz
Keyword(s):  
Recent Work ◽  
Contact Structures ◽  
Open Book ◽  
Open Book Decomposition ◽  
Dehn Twists ◽  
Left Handed ◽  
Closed Contact

In this note, we use the recent work of Honda-Kazez-Matić [3] to prove that a closed contact 3-manifold admitting a compatible open book decomposition with a nontrivial monodromy which can be presented as a product of left handed Dehn twists is overtwisted.

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