# Thurston norms of tunnel number-one manifolds

## TL;DR

This paper investigates the Thurston norms of certain 3-manifolds with first Betti number two, revealing that even restricted classes can have complex unit balls with many faces.

## Contribution

It characterizes the possible shapes of Thurston norm unit balls for 3-manifolds with specific algebraic properties, showing they can be arbitrarily complex.

## Key findings

- Existence of 3-manifolds with arbitrarily many faces in their Thurston norm unit ball
- Thurston norm unit balls can be highly complex even in restricted classes
- The study links algebraic presentations to geometric complexity

## Abstract

The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds $M$ with $b_1(M) = 2$, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are 3-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.