  An R_2-move is a homotopy of wrinkled fibrations which deforms images of
indefinite fold singularities like Reidemeister move of type II. Variants of
this move are contained in several important deformations of wrinkled
fibrations, flip and slip for example. In this paper, we first investigate how
monodromies are changed by this move. For a given fibration and its vanishing
cycles, we then give an algorithm to obtain vanishing cycles in one reference
fiber of a fibration, which is obtained by applying flip and slip to the
original fibration, in terms of mapping class groups. As an application of this
algorithm, we give several examples of diagrams which were introduced by
Williams to describe smooth 4-manifolds by simple closed curves of closed
surfaces.

Geometric Topology