The Irreducibility of Heegaard Splittings of Seifert Fibered Spaces

Moriah and Schultens have demonstrated that an irreducible Heegaard splitting of an orientable Seifert fibered space over an orientable base surface is either vertical or horizontal. In this paper it is determined precisely which vertical and horizontal splittings are irreducible. Let M be a Seifert fibered space which admits a horizontal splitting at the fiber f . If the genus of the horizontal splitting at f is less than the genus of the vertical splittings, its genus will be minimal and the splitting irreducible. Otherwise, this splitting will be irreducible if and only if the multiplicity of the fiber f is strictly greater than the least common multiple of the multiplicities of the other fibers. In particular, each Seifert fibered space possesses at most one irreducible horizontal splitting. The vertical splittings will be reducible if and only if M has a horizontal splitting with genus strictly less than the genus of the vertical splittings.