Essential open book foliation and fractional Dehn twist coefficient

2012/08/08 by Tetsuya Ito, Ito, Tetsuya, Keiko Kawamuro +1 · 2 citations
Mathematics · #57M50 (Primary) 53D35 #57R17 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1208.1559

openalex publication_date 2012/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce an essential open book foliation, a refinement of the open book foliation, 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' of overtwisted contact structures and a 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.

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

Cited by

• Overtwisted discs in planar open books
• Quasi right-veering braids and non-loose links

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