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

Let M be a connected, compact, orientable 3-manifold with b1(M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality ‖φ‖A ≤ ‖φ‖T between the Alexander norm on H(M,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 b1(M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings.