Abstract
We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the ‘gap’ between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen–Thurston types of the monodromies of their open book decompositions.
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Acknowledgments
The authors would like to thank Ken Baker, John Etnyre, Charlie Frohman, William Kazez and Dale Rolfsen for helpful conversations, and Jesse Hamer and Sam Brensinger for helping with English. They especially thank the referee for numerous instructive comments and for pointing out gaps and typos. TI was partially supported by JSPS Postdoctoral Fellowships for Research Abroad. KK was partially supported by NSF Grants DMS-1016138 and DMS-1206770.
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Ito, T., Kawamuro, K. Essential open book foliations and fractional Dehn twist coefficient. Geom Dedicata 187, 17–67 (2017). https://doi.org/10.1007/s10711-016-0188-7
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DOI: https://doi.org/10.1007/s10711-016-0188-7