Thurston norm via Fox calculus

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