That’s a good point, and I agree that part 1 probably follows from Maggy’s paper. The two-bridge case, which was proved by Tsuyoshi Kobayashi, uses a double sweep-out argument similar to the argument in Maggy’s paper, but it takes advantage of the fact that the two-bridge surface (a four-punctured sphere) has a Farey graph rather than a curve complex.

]]>This is a nice post and an interesting conjecture. It’s quite surprising to think that there could be 3-manifolds with many distinct minimal genus Heegaard splittings (this somehow seems very complicated), all of which are obtained by generic operations performed on a single high distance bridge surface (this somehow seems very uncomplicated).

With regards to part 1 of your conjecture, I think Maggy Tomova’s paper “Multiple bridge surfaces restrict knot distance” (http://arxiv.org/abs/math/0511139) might be helpful. She shows that if S and Q are two distinct irreducible bridge surfaces for a knot K, then the distance d(S) bounds a function of the Euler characteristic of Q from below. Although I haven’t read the paper closely, it appears that the result also holds when Q is a Heegaard surface. Thus, if S is a bridge surface, d(S) is above some threshold (2n for S an n-bridge sphere), and Q is a low genus Heegaard surface for the exterior of K, it should follow from Maggy’s main theorem that Q is one of the your candidate surfaces.

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