Presentations of the Mapping Class Group

The mapping class group is an important algebraic invariant of a surface. Presentations of this group have wide applications to low-dimensional topology. We explicitly construct Hatcher and Thurston's nite presentation with Dehn twist generators for genus one and two surfaces. We then extend Bene's chord slide presentation from surfaces with connected boundary to those with disconnected boundary. This presentation arises from studying a cell decomposition of Teichmuller space whose vertices are fatgraph decorations of surfaces. We can convert the resulting fatgraph presentation of the mapping class group to one with chord slide generators. This chord slide presentation has potential applications to computing bordered Heegaard Floer invariants for open books with disconnected binding