Flipping Heegaard splittings and minimal surfaces

We show that the number of genus $g$ embedded minimal surfaces in ${\mathbb{S}}^3$ tends to infinity as $g\to \mathrm{\infty }$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\to \mathrm{\infty }$, and arise from a two-parameter min-max scheme in lens spaces. More generally, by stabilizing and flipping Heegaard foliations we produce index at most $2$ minimal surfaces with controlled topological type in arbitrary Riemannian three-manifolds.