Turaev-Viro TQFT and the Rank versus Genus Conjecture

TL;DR

This paper develops a method to estimate the Heegaard genus of 3-manifolds using the Turaev-Viro TQFT derived from quantum groups, providing new bounds and counterexamples related to the rank versus genus conjecture.

Contribution

It introduces a novel approach to bound the Heegaard genus via unitary Turaev-Viro TQFT and applies computational tools to identify counterexamples to the conjecture.

Findings

Established a lower bound for Heegaard genus using Turaev-Viro TQFT.

Identified counterexamples to the rank versus genus conjecture with Regina software.

Connected TQFT properties to topological invariants of 3-manifolds.

Abstract

This paper presents a way to estimate the Heegaard genus of a $3$-manifold using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived from the modular category associated to the quantum group $U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev and Virelizier the corresponding TQFT is unitary. We modify a proof by Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT, and then use the software Regina to provide some known counterexamples to the rank versus genus conjecture.