Georgia Institute of Technology College of Sciences

In 1986, Thurston introduced a norm on the first cohomology of a 3-manifold $M$ and showed that it can be used to study which cohomology classes are induced by a fibration of $M$ over the circle. In 1998, McMullen introduced a norm on first cohomology that depends only on the Alexander polynomial and showed that it provides a lower bound for the Thurston norm. In this talk, we will introduce the Thurston and Alexander norms and explain why there is an inequality relating the two. To do this, we will define the Alexander polynomial in terms of elementary ideals, and we will use this perspective to understand how topological information is encoded in the exponents of the Alexander polynomial.