A REFINEMENT OF JOHNSON'S BOUNDING FOR THE STABLE GENERA OF HEEGAARD SPLITTINGS

For each integer k >= 2, Johnson gave a 3-manifold with Heegaard splittings of genera 2k and 2k - 1 such that any common stabilization of these two surfaces has genus at least 3k - 1. We modify his argument to produce a 3-manifold with two Heegaard splitings of genus 2k such that any common stabilization of them has genus at least 3k.