Suppose is a knot in with bridge number and bridge distance greater than . We show that there are at most distinct minimal-genus Heegaard splittings of . These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If has bridge distance at least , then two splittings from different families become equivalent only after stabilizations. Furthermore, we construct representatives of the isotopy classes of the minimal tunnel systems for corresponding to these Heegaard surfaces.
"Stabilizing Heegaard splittings of high-distance knots." Algebr. Geom. Topol. 16 (6) 3419 - 3443, 2016. https://doi.org/10.2140/agt.2016.16.3419