Parity condition for irreducibility of Heegaard splittings

TL;DR

This paper introduces a parity condition to determine the irreducibility of Heegaard splittings in irreducible manifolds, providing a new criterion that extends previous work and enables the construction of non-stabilized splittings via Dehn twists.

Contribution

It presents a novel parity condition for irreducibility of Heegaard splittings, expanding the tools available for analyzing 3-manifold decompositions.

Findings

Parity condition effectively detects irreducibility

Examples of non-stabilized splittings constructed by a single Dehn twist

Extension of Casson-Gordon's rectangle condition

Abstract

Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.