Narrow Results

Let P, Q be Heegaard surfaces of a closed orientable 3-manifold. In this paper, we introduce a method for giving an upper bound of (Hempel) distance of P by using the Reeb graph derived from a certain horizontal arc in the ambient space [0, 1] × [0, 1] of the Rubinstein–Scharlemann graphic derived from P and Q. This is a refinement of a part of Johnson's arguments used for determining stable genera required for flipping high distance Heegaard splittings.

In this paper, we define the rectangle condition for bridge decompositions of links in $S^3$ whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for an $n$-bridge decomposition can guarantee that the Hempel distance of the $n$-bridge decomposition is greater than or equal to two.