The Alexander Polynomial of a 3-Manifold and the Thurston Norm on Cohomology

Abstract

Let M be a connected, compact, orientable 3-manifold with (b_1(M)>1), whose boundary (if any) is a union of tori. Our main result is the inequality ({\parallel \phi \parallel}_A \le {\parallel \phi \parallel}_T) between the Alexander norm on (H^1(M,\mathbb{Z})), defined in terms of the Alexander polynomial, and the Thurston norm, defined in terms of the Euler characteristic of embedded surfaces. (A similar result holds when (b_1(M)=1).) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings.