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

A refinement of Johnson's bounding for the stable genera of Heegaard splittings