Parity criterion and Dehn twists for unstabilized Heegaard splittings

TL;DR

This paper introduces a parity condition for Heegaard diagrams to identify unstabilized splittings and demonstrates how Dehn twists can generate such splittings from existing ones.

Contribution

It provides a new parity criterion for unstabilized Heegaard splittings and shows how to construct them via Dehn twists, improving previous results.

Findings

Parity condition effectively detects unstabilized splittings

Dehn twists can produce unstabilized splittings from any given splitting

Enhanced understanding of Heegaard splitting construction methods

Abstract

We give a parity condition of a Heegaard diagram to show that it is unstabilized. This improves the result of [5]. As an application, we construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.