I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.)
I'm curious about a statement referring to: open books that cannot be further (de)stabilized. I'm having trouble finding a reasonable interpretation for this phrase. Moreover, I only know vaguely that the process of destabilization/deplumbing is the inverse process of that of stabilization (meaning that destabilization undoes the effects of stabilization.), but I don't have a clear, specific-enough description of the process of destabilization.
Let's see stabilization first: So, if we have an open book decomposition $(B, \pi)$ , where $B$ is the binding of the open book, i.e., $B$ is a fibered knot, and $\pi$ is the projection map so that $(M-B) \rightarrow S^1$ is a locally-trivial fiber bundle with fiber a surface $\Sigma$, we then "stabilize" by attaching a 2-D 1-handle and we choose a curve $C$ going once through the cocore of the handle. Then we do a positive Dehn twist about $C$ .The manifold $M':=M \cup$ 1-handle that results from this process of stabilization is homeomorphic to $M$, and the open book whose monodromy is composed with the Dehn twist still supports the original contact structure.
So, naively, it would seem we can destabilize and undo the effects of stabilization as in the last paragraph, by removing a 2-D 1-handle , removing a curve C going through the cocore once, and composing with a +/- Dehn twist about C ( the sign of the twist would depend on whether we are doing a +/- destabilization). But this seems too easy.
Equivalently, the process of stabilizing is equivalent to plumbing together the bindings (let's work on 3-manifolds for now ), and ending up with surfaces/fibers with transverse self-intersections, and an associated plumbing diagram, e.g: http://en.wikipedia.org/wiki/E8_manifold. But I don't see how to use this to define destabilization ( deplumbing?) , nor how to find associated results to destabilization, , e.g., obstructions to the process -- there must be some restrictions, since otherwise, we could always have genus-zero pages in our books.
Can anyone please explain this process and/or give a ref. for results associated with it? It would be great too, if the sources included plane-field invariants like d_3 , and how these invariants are affected by (de) stabilization.