An estimation of Hempel distance by using Reeb graph

TL;DR

This paper presents a new method to estimate the Hempel distance between Heegaard surfaces in 3-manifolds using Reeb graphs derived from Rubinstein-Scharlemann graphics, refining previous approaches for high-distance splittings.

Contribution

It introduces a novel approach to upper bound Hempel distance via Reeb graphs, enhancing the understanding of Heegaard splitting complexity.

Findings

Provides an upper bound estimate for Hempel distance

Refines Johnson's arguments on stable genera

Connects Reeb graphs with Heegaard surface analysis

Abstract

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 $[0,1]\times[0,1]$ 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.