On (disk, annulus) pairs of Heegaard splittings that intersect in one point

Bull. Korean Math. Soc. 2009 Vol. 46, No. 1, 99-105 Printed January 1, 2009

Jung Hoon Lee Korea Institute for Advanced Study

Abstract : Let $M=H_1\cup_S H_2$ 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 $M=H'_1\cup_{S'} H'_2$ by removing the neighborhood of $A$ from $H_2$ and attaching it to $H_1$, and show that $M=H'_1\cup_{S'} H'_2$ also has a $(D,A)$ pair with $|D\cap A|=1$.