On unstabilzed genus three critical Heegaard surfaces

TL;DR

This paper introduces a simplicial complex of weak reducing pairs for genus three Heegaard splittings, proving criticality under certain conditions and providing specific examples like $T^3$.

Contribution

It defines a new combinatorial structure for analyzing genus three Heegaard splittings and establishes conditions under which they are critical.

Findings

Unstabilized genus three Heegaard splittings can be critical.

The simplicial complex of weak reducing pairs has specific properties.

Standard genus three splitting of $T^3$ is shown to be critical.

Abstract

Let $M$ be a compact orientable irreducible 3-manifold and $H$ be an unstabilized genus three Heegaard splitting of $M$. In this article, we will define a simplicial complex of weak reducing pairs for $H$ and find several properties of this complex. Using this method, we will prove that an unstabilized Heegaard splitting of genus three is critical in a certain condition. In addition, we will show that the standard genus three Heegaard splitting for $T^3$ and the induced Heegaard splitting of the three component chain exterior by a certain tunnel system are critical as examples of the main theorem.