Operations on open book foliations

Abstract

Abstract. We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation.As applications, we study how open book foliations change under a stabilization of the open book. We also generalize Birman-Menasco's split/composite braid theorem: We show that closed braid representatives of a split (resp. composite) link in a certain open book can be converted to a split (resp. composite) closed braid by applying exchange moves finitely many times.

On a relation between the self-linking number and the braid index of closed braids in open books

“…As we have seen in [11,Example 2.20], there is a closed 1-braid α which is a transverse push-off of the boundary of an overtwisted disc (which we call a tranverse overtwisted disc), so sl( α) = 1. On the other hand, the meridian of a connected component of the binding is a closed 1-braid β with sl( β) = −1.…”

“…Among other things, foliation change and exchange move introduced in [3,4], and various observations and techniques developed in proving Markov Theorem Without Stabilization (MTWS) [6,7] and usual Markov theorem [5] play crucial roles. In our proof, we use an open book foliation machinery developed in [11,13,14,15] which is a generalization of the braid foliation.…”

“…We call them hyperbolic points. By isotopy fixing ∂A, A can be put so that it admits an open book foliation (see [11,Theorem 2.5…”

“…If F ob (A) contains at least one hyperbolic point, then we can decompose A as a union of regions whose interiors are disjoint [11,Proposition 3.11]. We call such a decomposition a region decomposition.…”

“…An interior exchange move, which was simply called an exchange move in [14], is an operation that removes four singular points. This operation may change the braid isotopy class of ∂A, but preserves the transverse link types.…”