The Turaev and Thurston norms

TL;DR

This paper explores the relationship between Thurston's and Turaev's norms on the first cohomology groups of 3-manifolds and 2-complexes, demonstrating their equivalence in the context of link exteriors in rational homology spheres.

Contribution

It establishes that Thurston's norm coincides with a variation of Turaev's norm for link exteriors in rational homology spheres, linking two important topological invariants.

Findings

Thurston and Turaev norms are equivalent for link exteriors in rational homology spheres.

The paper provides a variation of Turaev's norm applicable to 2-skeletons.

The results unify different approaches to measuring complexity in 3-manifolds.

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 a 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.