#### Volume 21, issue 6 (2017)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Thurston norm via Fox calculus

### Stefan Friedl, Kevin Schreve and Stephan Tillmann

Geometry & Topology 21 (2017) 3759–3784
##### 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.

##### Keywords
Thurston norm, $3$–manifold, Novikov ring, Fox calculus
##### Mathematical Subject Classification 2010
Primary: 20J05, 57M05, 57M27, 57R19