All:

I am looking for a proof of the result that any open book $(B, \pi)$ ; $B$ a fibered link , $\pi$ a map of $M^3-B \rightarrow S^1$ on a 3-manifold $M^3$ , so that $\pi$: $(M^3-B)$ fibers over $S^1$ , where the fibers $\pi^{-1}(\theta)$ are surfaces $\Sigma_{\theta}$ with $\partial \Sigma_{\theta}=B$ , can be obtained as an abstract open book (AOB). An AOB is a pair $(\Sigma, \phi)$ , for $\Sigma$ a compact surface with non-empty boundary , so that $\phi|_{\partial \Sigma}=Id$; $Id(x)=x$ , for $\Sigma$ and $\phi: \Sigma \rightarrow \Sigma$, a diffeomorphism called the monodromy of the AOB (i.e., $\phi$ is an element of the MCG of $\phi$).

The "actual" book is then obtained from the AOB by first doing the mapping torus $\Sigma_{\phi}$, and then filling-in a solid torus $D^2 \times S^1$ for each boundary component. The binding B then lives in the solid tori ( which are tubular neighborhoods of the respective components in the binding.) I would also appreciate a proof of the fact that the binding $B$ alone does not determine the open book, i.e., $M^3 -B$ may be fibered over $S^1$ in non-isotopic ways.

I'm specially interested on how to determine the monodromy once an "actual" , i.e., non-abstract, open book is given, i.e., given the data $(B, \pi)$ for $M^3$ , how do I obtain the needed $\phi$ ?

Thanks for any help, refs, etc.