Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})$ and the open book decomposition induces a contact structure $\xi$ on $Y$. Is there any criterion for tightness/overtwistedness of $\xi$ based on the matrix $\phi$?



Tightness/Overtwistedness of genus one open book decomposition
[Honda–Kazez–Matić](https://arxiv.org/pdf/math/0609734.pdf) and [Baldwin](https://arxiv.org/pdf/math/0604580.pdf) proved that tight here is equivalent to right-veering, so you can sit down and work that out. The former paper shows that reducible monodromies always give you a tight contact structure, and periodic ones give you a tight contact structure if and only if the monodromy's a product of positive Dehn twists.


To save yourself some time, Baldwin's paper has this worked out up to conjugacy (that is, conjugate to a certain factorization). Look at his Theorem 4.3, using notation from his Theorem 2.6. If you just want pseudo-Anosov monodromies, his Theorem 1.2 is a go-to.


