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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 and on about this story for quite some time, but I think that I'm better off referring to Etnyre's classical Lectures on open book decompositions and contact structures.

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), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

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 and here). Some of their examples are of genus two.

The question for genus one open books with disconnected boundary is - as far as I know - still open.

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 and on about this story for quite some time, but I think that I'm better off referring to Etnyre's classical Lectures on open book decompositions and contact structures.

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), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

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 and here). Some of their examples are of genus two.

The question for genus one open books with disconnected boundary is - as far as I know - still open.

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.

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), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

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 and here). 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.

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 and on about this story for quite some time, but I think that I'm better off referring to Etnyre's classical Lectures on open book decompositions and contact structures.

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), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

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 and here). Some of their examples are of genus two.

The question for genus one open books with disconnected boundary is - as far as I know - still open.

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 $d_3 = +1/2$, if I recall correctly) and the resulting contact structure is overtwisted, with the same Chern class as $\theta$ but with a different $d_3$ (shifted by $\pm1/2$ -- I need to check this).

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), 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 Etnyre's Lectures on open book decompositions and contact structures contain a lot of information.

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 organise a lot of results, proofs and references). For planar surfaces and genus-1 surfaces with connected boundary, there are even stronger, very recent statements which are not contained in the book.

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.

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), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

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 and here). 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.