Regarding your last, non-highlighted question: the operation you describe is called Hopf stabilisation, and it correspond to doing a connected sum of $\theta$ with a contact 3-sphere (the binding is connected-summed to a Hopf link -- whence the name). Therefore, on the manifold level this doesn't change anything. Let's see what happens to the contact structure. If the Dehn twist along $C$ is negative (or left-handed), the Hopf link is negative (_i.e._ the two components have linking number -1), and the contact structure on the 3-sphere is overtwisted (with $d_3 = +1/2$) and the resulting contact structure $\theta'$ is overtwisted, with the same Chern class as $\theta$ but with a different $d_3$ (namely, $d_3(\theta') = d_3(\theta)+1$). If the Dehn twist along $C$ is positive (right-handed), the Hopf link is positive (_i.e._ the two components have linking number +1), then the contact structure on the 3-sphere is the standard one (the only tight one, which has $d_3 = -1/2$), and this doesn't affect $\theta$ at all. I think that this is also called a Giroux stabilisation. Giroux proved that two contact structures given by two open books $OB_1 = (S_1,f_1)$, $OB_2 = (S_2,f_2)$ are isotopic if and only if $OB_1$ and $OB_2$ are stably equivalent, that is if and only if they have a common (positive) stabilisation. I could go on an on about this story for quite some time, but I think that I'm better of referring to Etnyre's classical [_Lectures on open book decompositions and contact structures_][1]. --- Regarding Stein fillability, a contact structures is Stein fillable if and only if it has a supporting open book whose monodromy can be factorised as the product of positive Dehn twists. If you want to know more about fillability and Stein surfaces, Ozbagci and Stipsicz's _Surgery on contact 3-manifolds and Stein surfaces_ makes for a really pleasant read (in addition to organising a lot of results, proofs and references). The converse to the positivity statement above is true in some cases. First Wendl proved that if a _planar_ open book supports a Stein fillable contact structure, then the monodromy is a product of right-handed Dehn twists (see [here][4]), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see [here][5]). It is not true in general: in the meantime, Wand, and Baker, Etnyre and Van Horn-Morris found example of monodromies that were not isotopic to a product of right-handed Dehn twist, but whose associated open books support Stein fillable contact structures (see [here][2] and [here][3]). Some of their examples are of genus 2. The question for genus-1 open books with disconnected boundary is - as far as I know - still open. [1]: http://people.math.gatech.edu/~etnyre/preprints/oblec.html [2]: http://arxiv.org/abs/0910.5691 [3]: http://arxiv.org/abs/1005.1978 [4]: http://arxiv.org/abs/0806.3193 [5]: http://arxiv.org/abs/1304.1299