The irreducibility of Heegaard splittings of Seifert fibered spaces

Moriah and Schultens have demonstrated that an irreducible Heegaard splitti ng of an orientable Seifert fibered space over an orientable base surface i s either vertical or horizontal. In this paper it is determined precisely w hich vertical and horizontal splittings are irreducible. Let M be a Seifert fibered space which admits a horizontal splitting at the fiber f. If the g enus of the horizontal splitting at f is less than the genus of the vertica l splittings, its genus will be minimal and the splitting irreducible. Othe rwise, this splitting will be irreducible if and only if the multiplicity o f the fiber f is strictly greater than the least common multiple of the mul tiplicities of the other fibers. In particular, each Seifert fibered space possesses at most one irreducible horizontal splitting. The vertical splitt ings will be reducible if and only if M has a horizontal splitting with gen us strictly less than the genus of the vertical splittings.