Quantum invariants can provide sharp Heegaard genus bounds

TL;DR

This paper shows that quantum invariants, specifically Reshetikhin-Turaev invariants, can give precise lower bounds on the Heegaard genus of certain three-manifolds, surpassing bounds from fundamental group rank.

Contribution

It introduces a method to use quantum invariants for sharp Heegaard genus bounds, improving upon traditional algebraic bounds.

Findings

Quantum invariants provide sharper bounds than fundamental group rank.

Reshetikhin-Turaev invariants can be used to estimate Heegaard genus.

Examples demonstrate the effectiveness of quantum invariants in topology.

Abstract

Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.