We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken –manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer there are –manifolds containing an essential torus and a Heegaard splitting such that the torus and splitting surface must intersect in at least simple closed curves. These give the first examples of lower bounds on the minimum number of curves of intersection between an essential surface and a Heegaard surface that are greater than one.
"Stabilization, amalgamation and curves of intersection of Heegaard splittings." Algebr. Geom. Topol. 9 (2) 811 - 832, 2009. https://doi.org/10.2140/agt.2009.9.811