Visualizing Overtwisted Discs in Open Books

Abstract

We give an alternative proof of a theorem of Honda-Kazez-Matic 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. In (15), we have introduced open book foliations and their basic machinery by using that of braid foliations (2, 3, 4, 5, 6, 7, 8, 9) and showed applications of open book foliations including a self-linking number formula of general closed braids. In (16) we study the geometric structure of a 3-manifold by using open book foliations. In this note we study more applications of open book foliations. We will assume the readers are familiar with the definition and basic machinery of open book foliations in (15). One of the features of open book foliations is that one can visualize how surfaces are embedded with respect to general open books. In this paper we use this feature to illustrate overtwisted discs and give constructive methods to detect overtwisted contact structures. We first give an alternative proof of a tightness criterion theorem by Honda, Kazez and Matic (14): If an open book is not right-veering then it supports an overtwisted contact structure. The converse does not hold: In fact, Honda, Kazez and Matic (14) show that if a contact structureis supported by a non-right veering open book (S,�), by applying positive stabilizations to (S,�) one can find a right-veering open book ( b S, b �) that supports �. We