# On keen weakly reducbile Heegaard splittings

## TL;DR

This paper introduces a construction method for keen weakly reducible Heegaard splittings of arbitrary genus (except 2) and explores conditions under which critical Heegaard splittings can be obtained.

## Contribution

It provides a new construction technique for keen weakly reducible Heegaard splittings of arbitrary genus and discusses how to obtain critical splittings by modifying conditions.

## Key findings

- Constructed keen weakly reducible Heegaard splittings for genus ≥ 3
- Identified conditions leading to critical Heegaard splittings
- Extended understanding of the structure of weakly reducible splittings

## Abstract

A Heegaard splitting which admits a unique pair of disjoint compression disks on distinct sides is said to be keen weakly reducible. This paper provides an construction of keen weakly reducible Heegaard splittings of arbitrary genus except 2. Furthermore, critical Heegaard splittings may yield if we change some conditions.