The Turaev and Thurston norms

Stefan Friedl, Daniel S. Silver and Susan G. Williams

Vol. 284 (2016), No. 2, 365–382

DOI: 10.2140/pjm.2016.284.365

Abstract

In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a $3$-manifold. Inspired by this definition, Turaev introduced in 2002 an analogous norm on the first cohomology group of a finite 2-complex. We show that if $N$ is the exterior of a link in a rational homology sphere, then the Thurston norm agrees with a suitable variation of Turaev’s norm defined on any 2-skeleton of $N$.

Keywords

Turaev norm, Thurston norm, links, Alexander polynomials

Mathematical Subject Classification 2010

Primary: 57M27

Milestones

Received: 30 July 2015

Revised: 23 February 2016

Accepted: 23 March 2016

Published: 30 August 2016