Elliptic open books on torus bundles over the circle

Abstract

ABSTRACT. As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.

Tight small Seifert fibered manifolds with $e_0=-2$

“…When i = 0, the open book is compatible with the unique Stein fillable contact structure ξ 0 . This much is a slight modification of the picture in Van Horn-Morris thesis [33,Theorem 4.3.1.d] (see also [2,Table 2]). Adding the segment of words as in the lower part of the open book i times corresponds to increasing the Giroux torsion by i.…”

Partial twists and exotic Stein fillings

“…Van Horn-Morris [45] and subsequently Etgü [24] constructed a genus one open book of the Stein fillable contact structure on the 3-torus T 3 . Note that T 3 has a unique Stein fillable contact structure up to contactomorphism ( [19]).…”

“…Definition 4.1. Let Φ be the element of Aut(S, ∂S) defined by [45] and Etgü [24]). The open book (S, Φ) is compatible with the unique Stein fillable contact structure on T 3 .…”

“…Similarly to the proofs of this theorem in [45] and [24], we factorize Φ into a product of right handed Dehn twists. We moreover give infinitely many factorizations.…”