Open books supporting overtwisted contact structures and Stallings twist

Details Open books supporting overtwisted contact structures and Stallings twist Open books supporting overtwisted contact structures and Stallings twist

2006-08-15

arxiv.org/abs/math/0608371v3

Mathematics

Geometric Topology

11 pages, 12 figures

Scientific paper

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.

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