Thurston Polytopes

Abstract

We explore a 3-manifold and link invariant called the Thurston norm which provides a deep understanding of the submanifold structure of a 3-manifold. Recently, methods to compute the Thurston norm ball (a symmetric rational polytope) have been developed, providing a doorway through which we can hope to understand more about this invariant. In particular we use these techniques in order to find patterns in these Thurston norm balls which give rise to new conjectures. We also showcase some existing literature in the field to highlight relationships between ∥ · ∥T and other properties/invariants of manifolds and links. This work embarks on a journey through low dimensional topology making stops in fields as diverse as combinatorial group theory, Floer homology, and hyperbolic geometry. In this way, we hope to convince the reader that this invariant is well worth study.