Plumbing constructions of connected divides and the Milnor fibers of plane curve singularities

A divide is the image of a generic, relative immersion of a finite number of copies of the unit interval or the unit circle into the unit disk. N. A'Campo defined for each connected divide a link in S³ and proved that the link is fibered. In the present paper we show that the fiber surface of the fibration of a connected divide can be obtained from a disk by a successive plumbing of a finite number of positive Hopf bands. In particular, this gives us a geometric understanding of plumbing constructions of the Milnor fibers of isolated, complex plane curve singularities in terms of certain replacements of the curves of their real modifications.