Visualizing Overtwisted Discs in Open Books

Abstract

We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.

Coverings of Open Books

“…We prove the assertion (1). Applying the proof of Theorem 4.1 in [16] we can construct a transverse overtwisted disk in the open book (S, φ). By the definition every transverse overtwisted disk has the self-linking number 1, that is, the Bennequin-Eliashberg inequality [5] is violated.…”

Quasi-right-veering braids and nonloose links

“…We now prove our main theorem: Our proof of Theorem 4.1 is a generalization of the proof of [19,Theorem 2.4]. We may assume that the readers are familiar with basic definitions and properties of open book foliations that can be found in [18,20,21].…”

“…We explicitly construct a transverse overtwisted disk D trans in M (S,ϕ) \ L by giving its movie presentation. A similar construction can be found in [19]. Here, a transverse overtwisted disk (see [18,Definition 4.1] for the precise definition) is a disk admitting a certain type of open book foliation and is bounded by a transverse push-off of a usual overtwisted disk.…”

“…Since [ϕ L ](α) = α 0 , the slices D trans ∩ S 1 and D trans ∩ S 0 of D trans can be identified under the distinguished monodromy ϕ L . In other words the movie presentation gives rise to an embedded surface in M (S,ϕ) \ L. The construction tells us that the surface is topologically a disk, and moreover it is a transverse overtwisted disk (see [19]). Let γ be the isotopy class of a co-core of the attached 1-handle S \ S N .…”

Operations on open book foliations

“…This is a sequel of the papers [19,20,21] on open book foliations in which techniques to study the topology and contact structures of 3-manifolds are developed. The idea of an open book foliation originally came from the works of Bennequin [1] and Birman and Manasco [3,4,5,6,7,8,11,12].…”