Full Lutz twist along the binding of an open book

Abstract

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