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 $\phi$?
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2 Answers
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Honda–Kazez–Matić and Baldwin 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.
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A general criterion for tightness/overtwistedness of the contact structure in terms of the page and of the monodromy of the open book was given by Andy Wand: "Tightness is preserved by Legendrian surgery", Annals of Mathematics 182 (2015), 1-16
