Thurston norm via Fox calculus

Abstract

In 1976 Thurston associated to a $3$–manifold $N$ a marked polytope in $H_1\left(N;ℝ\right)$, which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in $H^1\left(N;ℝ\right)$. Recently the first and third authors associated to a presentation $\pi$ with two generators and one relator a marked polytope in $H_1\left(\pi ;ℝ\right)$ and showed that it determines the Bieri–Neumann–Strebel invariant of $\pi$. We show that if the fundamental group of a $3$–manifold $N$ admits such a presentation $\pi$, then the corresponding marked polytopes in $H_1\left(N;ℝ\right)=H_1\left(\pi ;ℝ\right)$ agree.