Thurston norm via Fox calculus

TL;DR

This paper connects Thurston's polytope for 3-manifolds with a Fox calculus-based polytope derived from a specific group presentation, showing they coincide under certain conditions.

Contribution

It demonstrates that for 3-manifolds with a two-generator, one-relator fundamental group, the Thurston polytope and the Fox calculus polytope are equivalent.

Findings

Thurston polytope and Fox calculus polytope coincide for certain 3-manifolds.

The result links geometric and algebraic invariants of 3-manifolds.

Provides a new method to compute Thurston norms using Fox calculus.

Abstract

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.