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.

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.