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Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\theta)$ on $\mathbb{R}^{3}$, where $(r, \theta, z)$ are cylindrical coordinates on $\mathbb{R}^{3}$.

Greetings, Mortel

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    $\begingroup$ I'm confused, surely you would find examples when searching the literature? Here is a very useful OT contact form on $S^1\times S^2$, it pops up when studying 4-manifolds that need not be symplectic and also in gauge theory: $\lambda=(1-3\cos^2\theta)dt+\sqrt{6}\cos\theta\sin^2\theta\,d\varphi$, where $(\theta,\varphi)$ are the 2-spherical coordinates such that $0\le\theta\le\pi$. $\endgroup$ – Chris Gerig Jun 28 '18 at 17:22
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I take your question as asking for an explicit formula on a closed $3$-manifold? In this case, the most interesting example I know is the following:

The standard contact form on the $3$-sphere $\mathbb{S}^3$ in $\mathbb{C}^2$ is

$$ \alpha_0 := i\, \sum_{j=1}^2 (z_j\,d\bar z_j - \bar z_j\, dz_j) \;.$$

Choosing the function $f(z_1,z_2) = z_1^2 + z_2^2$, one can verify that $\alpha_0$ is adapted to the open book $(K,\vartheta)$ with binding $K = \{f=0\}$ and fibration over the circle given by $\vartheta = f/|f|$. The page of this open book is $T^*\mathbb{S}^1$ and the monodromy is a positive Dehn twist.

If we now subtract the term $i\,(f\,d\bar f - \bar f\, df)$ from $\alpha_0$, we obtain $$ \alpha_- := i\, \sum_{j=1}^2 (z_j\,d\bar z_j - \bar z_j\, dz_j) - i\,(f\,d\bar f - \bar f\, df) \;,$$

which is still a contact form, it is still supported by an open book with binding $K=f^{-1}(0)$ and with identical pages, but the induced orientation of the $3$-sphere is now different, and thus also the coorientation of the pages. The open book is then $(K,\bar\vartheta = \bar f/|f|)$, i.e., it has the same pages, but its monodromy is a negative Dehn twist.

You can also find the overtwisted disk explicitly by intersecting the $3$-sphere with a $3$-dimensional subspace of $\mathbb{C}^2$: The binding $K$ of the open book is given by the two circles $$K = \bigl\{(z,- i z)\in \mathbb{S}^3\bigr\}\sqcup \bigl\{(z,+ i z)\in \mathbb{S}^3\bigr\}\;.$$ Let $E\subset \mathbb{C}^2$ be the $3$-space that is spanned (over $\mathbb{R}$) by the complex line $(z,-iz)$ and the real line $(t,it)$. The intersection of $E$ with $\mathbb{S}^3$ is clearly a $2$-sphere. Its equator is one of the components of the binding; the north and south poles are $\pm(1,i)$, that is, the poles are the intersection of $\mathbb{S}^2$ with the second binding component.

Parametrize the $2$-sphere by $$\Phi\colon (\varphi, \vartheta) \mapsto \frac{1}{\sqrt{2}}\, \sin \vartheta \cdot \begin{pmatrix}e^{i\varphi} \\ -ie^{i\varphi}\end{pmatrix} + \frac{1}{\sqrt{2}}\,\cos\vartheta\cdot \begin{pmatrix}1 \\i \end{pmatrix}$$ with $\varphi\in[0,2\pi]$ and $\vartheta\in [0,\pi]$. The pull-back of $\alpha_0$ with this map is: $$ \Phi^*\alpha_0 = 2 \sin^2 \vartheta\, d\varphi $$ so that the characteristic foliation for $\alpha_0$ is just that of a $2$-sphere with segments running down from the north pole to the south pole.

We obtain $f\bigl(\Phi(\varphi, \vartheta)\bigr) = 2\, \cos\vartheta\, \sin\vartheta \, e^{i\varphi}$, so that $\Phi^*(f\,d\bar f - \bar f\, df) = -8i\,\cos^2\vartheta\, \sin^2\vartheta\, d\varphi$ and thus $$\Phi^*\alpha_- = 2\sin^2\vartheta\, \bigl(1 -4\,\cos^2\vartheta \bigr)\, d\varphi \;.$$ This shows that the $2$-sphere contains an overtwisted disk in the northern hemisphere and a second one in the southern hemisphere, because $\Phi^*\alpha_-$ becomes singular along the two circles $\cos\vartheta = \pm 1/2$ (ie $\vartheta = \pi/3$ or $\vartheta = 2\pi/3$).

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Not sure if I understand what you mean by non-trivial (first of all, $\mathbb{R}^3$ is not closed...) but Eliashberg proved that there is a distinct isotopy class of overtwisted contact structures on a closed 3-manifold for every homotopy class of tangent 2-plane fields. Thus there is e.g. an infinite set of isotopy classes of OT contact structures on $S^3$. See also chapter 4 of Geiges's book "Introduction to Contact Topology"; 4.7 discusses Eliashberg's result and 4.2 gives some helpful information about homotopy classes of plane fields.

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