Alexander and Thurston norms of graph links

TL;DR

This paper proves that Alexander and Thurston norms are identical for all irreducible Eisenbud-Neumann graph links in homology 3-spheres and extends this equivalence to broader classes of links in S^3, using JSJ decomposition and fibration obstructions.

Contribution

It establishes the equivalence of Alexander and Thurston norms for all irreducible Eisenbud-Neumann graph links and broader classes of links in S^3, linking geometric and algebraic invariants.

Findings

Alexander and Thurston norms coincide for all irreducible Eisenbud-Neumann graph links.

The norms are equal for all links in S^3 under certain JSJ decomposition conditions.

Every facet of the reduced Thurston norm unit ball of a graph link is a fibered facet.

Abstract

We show that the Alexander and Thurston norms are the same for all irreducible Eisenbud-Neumann graph links in homology 3-spheres. These are the links obtained by splicing Seifert links in homology 3-spheres together along tori. By combining this result with previous results, we prove that the two norms coincide for all links in S^3 if either of the following two conditions are met; the link is a graph link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all Seifert-fibered, or the link is alternating and not a (2,n)-torus link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all hyperbolic. We use the E-N obstructions to fibrations for graph links together with the Thurston cone theorem on link fibrations to deduce that every facet of the reduced Thurston norm unit ball of a graph link is a fibered facet.