We prove that transverse links in any contact manifold (M,ξ) can be realized as a sub-binding of a compatible open book decomposition. We define the support genus of a transverse link and prove that the support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it. Next, we find a relationship between the support genus of a transverse link and its Legendrian approximation.

I would like to thank John Etnyre and Shea Vela-Vick for several enlightening conversations. I would also like to thank the anonymous referee for their helpful suggestions and insightful comments. This research is partially supported by NSF Grant 1907654 and the SFB/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics, funded by the Deutsche Forschungsgemeinschaff (Project- ID 281071066-TRR 191)".