Full Lutz twist along the binding of an open book

TL;DR

This paper provides an explicit construction of an open book decomposition compatible with a contact structure obtained by performing a full Lutz twist along a binding component, using a local modification near the binding.

Contribution

It introduces a concrete method to modify open books locally to realize contact structures after a full Lutz twist along a binding.

Findings

Explicit open book $( ext{ } ext{ } ext{ } ext{ })$ construction for Lutz-twisted contact structures.

Local modification near the binding component to achieve the new open book.

Applicable to compatible open books of contact 3-manifolds.

Abstract

Let $T$ denote a binding component of an open book $(\Sigma, \phi)$ compatible with a closed contact 3-manifold $(M, \xi)$. We describe an explicit open book $(\Sigma', \phi')$ compatible with $(M, \zeta)$, where $\zeta$ is the contact structure obtained from $\xi$ by performing a full Lutz twist along $T$. Here, $(\Sigma', \phi')$ is obtained from $(\Sigma, \phi)$ by a \emph{local} modification near the binding.