  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.
