Thurston norm via Fox calculus

Friedl, Stefan; Schreve, Kevin; Tillmann, Stephan

doi:10.2140/gt.2017.21.3759

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Thurston norm via Fox calculus

Stefan Friedl, Kevin Schreve and Stephan Tillmann

Geometry & Topology 21 (2017) 3759–3784

DOI: 10.2140/gt.2017.21.3759

Abstract

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

Keywords

Thurston norm, $3$–manifold, Novikov ring, Fox calculus

Mathematical Subject Classification 2010

Primary: 20J05, 57M05, 57M27, 57R19

References

Publication

Received: 5 June 2016

Revised: 25 October 2016

Accepted: 29 November 2016

Published: 31 August 2017

Proposed: Ian Agol

Seconded: Peter Teichner, András I. Stipsicz

Authors

Stefan Friedl
Fakultät für Mathematik Universität Regensburg Regensburg Germany

Kevin Schreve
Department of Mathematics University of Michigan Ann Arbor, MI United States

Stephan Tillmann
School of Mathematics and Statistics The University of Sydney Sydney, NSW Australia

Volume 21, issue 6 (2017)