Narrow Results

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

Loi–Piergallini and Akbulut–Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the disk D² with bounded fibers. In this note we give a more intrinsic alternative proof of this result.

This work is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph [7]. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close control on the way in which the whole family of fibre surfaces meet the family of discs spanning the braid axis. We show how such a braid can be presented naturally as a word in the 'band generators' of the braid group discussed by Birman, Ko and Lee [1] in their recent account of the band presentation of the braid groups. In this context we are able to convert the conditions for mutual braiding into the existence of a suitable sequence of band relations and other moves on the braid word, and thus derive a combinatorial method for deciding whether a braid is mutually braided.