Multiple genus 2 Heegaard splittings: a missed case

TL;DR

This paper investigates overlooked cases of multiple genus 2 Heegaard splittings in closed orientable 3-manifolds, presenting new examples that challenge previous assumptions about their properties and stabilization behavior.

Contribution

It introduces new examples of 3-manifolds with multiple genus 2 Heegaard splittings, expanding understanding of their properties and stabilization issues.

Findings

Some examples have Hempel distance 3

Not all examples share properties from previous studies

Single stabilization may not always make splittings isotopic

Abstract

A gap in a paper of Rubinstein-Scharlemann is explored: new examples are found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard splittings. Properties common to all the examples in the original paper are not universally shared by the new examples: some of the new examples have Hempel distance 3, and it is not clear that a single stabilization always makes the multiple splittings isotopic.