On (disk, annulus) pairs of Heegaard splittings that intersect in one point
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Let M=H_1∪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∪{S'} H'_2 by removing the neighborhood of A from H_2 and attaching it to H_1, and show that M=H'_1∪{S'} H'_2 also has a (D,A) pair with │D∩A│=1.
Abstract
Let M=H_1∪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∪{S'} H'_2 by removing the neighborhood of A from H_2 and attaching it to H_1, and show that M=H'_1∪{S'} H'_2 also has a (D,A) pair with │D∩A│=1.
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