all: I want to know how to find out , hopefully in coordinates, (but I'll take what's available) , the description of the projection map in an abstract open-book decomposition.

Open book decompositions come in two main types (see below for more): "actual ones" (just called OBD: Open Book Deomps) and the "abstract ones" (AOBD) . The first is explicitly given, the second consists of a pair (Surface, Self-Diffeomorphism) that gives rise to one, i.e., to an actual AOB.

http://en.wikipedia.org/wiki/Open_book_decomposition

An Open Book Decomposition (OBD) of a 3-manifold $M^3$ is a pair $(B, \pi)$ , where B is a link, a.k.a., the binding of the OB , which is a submanifold of $M^3$ (a codimension-2 submanifold of $M^3$ ),and $\pi$ is a map so that ($M^3 - B)\rightarrow S^1$ is a fibrarion, i.e., a locally-trivial bundle,in which the fibers are surfaces whose boundary is the binding set B, while the restriction of $\pi$ to B is the angular coordinate; since B is a submanifold of M, it admits a tubular neighborhood $N(B)=B\times D^2$ , where B sits as ${0}\times D^2$. The abstract type is a pair (S,f); S is a compact, oriented surface with non-empty boundary , and f is a self-diffeomorphism that equals Id near $\partial S$, i.e., an element of MCG(S) (mapping-class group), and the pair gives rise to an AOB by the construction of a3-manifold $N^3$ defined by:

$N^3:= N_f \cup (S^1 \times D^2)$

Where $N_f$ is the mapping torus of S by f, and we glue-in solid tori along each boundary component (remember that $f|_{ \partial S}=Id$ )

And we get an OBD from $N^3$ in which the link/binding B is the core of the solid tori,the the fiber surfaces are the disks along the core. I'm just curious: does anyone know a nice expression for the projection map in an AOB $N^3$?

Thanks.