Journal of the Mathematical Society of Japan

Open books supporting overtwisted contact structures and the Stallings twist

Ryosuke YAMAMOTO

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Abstract

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 equivalent up to positive stabilization to the existence of an overtwisted disk in the contact manifold supported by the open book. We also show a criterion for overtwistedness using a certain arc properly embedded in the fiber surface, which is an extension of Goodman's one.

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 3 (2007), 751-761.

Dates
First available in Project Euclid: 5 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191591856

Digital Object Identifier
doi:10.2969/jmsj/05930751

Mathematical Reviews number (MathSciNet)
MR2344826

Zentralblatt MATH identifier
1139.57025

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
contact structure open book decomposition Stallings twist

Citation

YAMAMOTO, Ryosuke. Open books supporting overtwisted contact structures and the Stallings twist. J. Math. Soc. Japan 59 (2007), no. 3, 751--761. doi:10.2969/jmsj/05930751. https://projecteuclid.org/euclid.jmsj/1191591856


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