Suppose K is a knot in S3 with bridge number n and bridge distance
greater than 2n. We show that there are at most (n2nβ) distinct
minimal genus Heegaard splittings of S3βΞ·(K). These splittings
can be divided into two families. Two splittings from the same family become
equivalent after at most one stabilization. If K has bridge distance at least
4n, then two splittings from different families become equivalent only after
nβ1 stabilizations. Further, we construct representatives of the isotopy
classes of the minimal tunnel systems for K corresponding to these Heegaard
surfaces.Comment: 19 pages, 8 figure