  Non-isotopic Heegaard splittings of non-minimal genus were known previously
only for very special 3-manifolds. We show in this paper that they are in fact
a wide spread phenomenon in 3-manifold theory: We exhibit a large class of
knots and manifolds obtained by Dehn surgery on these knots which admit such
splittings. Many of the manifolds have irreducible Heegaard splittings of
arbitrary large genus. All these splittings are horizontal and are isotopic,
after one stabilization, to a multiple stabilization of certain canonical low
genus vertical Heegaard splittings.
