Journal of the Mathematical Society of Japan

Open books supporting overtwisted contact structures and the Stallings twist


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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

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

First available in Project Euclid: 5 October 2007

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Zentralblatt MATH identifier

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}

contact structure open book decomposition Stallings twist


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.

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