Modification rule of monodromies in R_2-move
  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. 
Kenta Hayano
math.GT