Compatible relative open books on relative contact pairs via generalized square bridge diagrams

Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact 3-manifold which results from a contact \((\pm 1)\)-surgery on a Legendrian link in the standard contact 3-sphere. In this article, we introduce the “generalized square bridge position” for a Legendrian link in the standard contact 5-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs.