ON (DISK, ANNULUS) PAIRS OF HEEGAARD SPLITTINGS THAT INTERSECT IN ONE POINT

Let $M=H_1{\cup}_SH_2$ be a Heegaard splitting of a 3-manifold M, D be an essential disk in $H_1$ and A be an essential annulus in $H_2$. Suppose D and A intersect in one point. First, we show that a Heegaard splitting admitting such a (D, A) pair satisfies the disjoint curve property, yet there are infinitely many examples of strongly irreducible Heegaard splittings with such (D, A) pairs. In the second half, we obtain another Heegaard splitting $M=H by removing the neighborhood of A from $H_2$ and attaching it to $H_1$, and show that $M=H also has a (D, A) pair with $|D{\cap}A|=1$.