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Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page at one point, so that $K$ is a transversal knot for the contact structure $\xi$ compatible with the open book. Thus, the monodromy of the open book, with this section, is an element $\phi \in \mathrm{Mod}_{g,b}^1$, the mapping class group of a genus-$g$ surface with $b\geq 1$ boundary components and with one marked point. The question is how to represent, in terms of the page and monodromy, an open book representing the Lutz twist of $\xi$ along $K$.

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    $\begingroup$ If there were an algorithm to turn your presentation of $K$ into the presentation of a Legendrian approximation $L$ lying (as an essential curve) on the page of an open book, then Ding--Geiges--Stipsicz tell you how to proceed (see here: arxiv.org/pdf/math/0401338.pdf ). (Both stabilisations and +1-surgeries are easy to represent on a page, then.) $\endgroup$ – Marco Golla Dec 27 '17 at 15:23

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