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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 H1(N; ), which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in H1(N; ). Recently the first and third authors associated to a presentation π with two generators and one relator a marked polytope in H1(π; ) 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 H1(N; ) = H1(π; ) agree.

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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