On the weak reducing pairs in critical Heegaard splitting

TL;DR

This paper investigates the properties of weak reducing pairs in critical Heegaard splittings, providing insights into their structure and a necessary condition for a surface to be classified as critical.

Contribution

It introduces new properties of weak reducing pairs in critical Heegaard splittings and establishes a necessary condition for the criticality of the Heegaard surface.

Findings

Identifies properties of weak reducing pairs in critical splittings.

Provides a necessary condition for a Heegaard surface to be critical.

Enhances understanding of the structure of critical Heegaard splittings.

Abstract

A weak reducing pair in a Heegaard splitting M = V \cup_S W is a pair of disjoint essential disks D \in V and E \in W. The weakly reducible Heegaard splitting contains at least one weak reducing pair. Critical Heegaard splitting is a special case of weakly reducible Heegaard splitting which contains at least two weak reducing pairs satisfying some special conditions. In this paper, we discuss the properties of weak reducing pairs in a critical Heegaard splitting and give a necessary condition for Heegaard surface to be critical.