A contact invariant in sutured monopole homology

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Mati\'c's contact invariant in sutured Heegaard Floer homology (SFH). In the process of defining our invariant, we construct maps on SHM associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Mati\'c in SFH. We use these maps to establish a bypass exact triangle in SHM analogous to Honda's in SFH. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in [1] to define a contact invariant in the instanton Floer setting.