Homeomorphisms between pairs of genus two handlebodies and separating circles in their boundaries
A known result in low-dimensional topology is that every incompressible surface properly embedded in a genus two handlebody is boundary compressible. This result produces a pair consisting of a genus two handlebody and a separating curve in its boundary. We study particular examples of genus two handlebodies and essential simple closed curves which separate their boundaries. The aim of this paper is to give our own construction of separating curves on the boundary of genus two handlebodies and compare them with previously studied pairs. These constructions involve taking 2n arcs on either side of a waist disk of a genus two handlebody and defining a mapping which takes the endpoints of one side and connects them to the other side. Before studying homeomorphisms with known pairs, we must first state and prove restrictions on the mapping of the arcs in our construction. Only particular cases of mappings will result in a single circle component and even fewer cases will result in homeomorphisms with previously studied pairs.^
Vazquez, Daniel, "Homeomorphisms between pairs of genus two handlebodies and separating circles in their boundaries" (2014). ETD Collection for University of Texas, El Paso. AAI1564703.