Complexity of 3–manifolds Geometric and Topological Perspectives

Abstract

Minimal triangulations of 3--dimensional manifolds offer key insights to topological and geometric properties of the underlying manifold. We study minimal triangulations through two lenses -- one geometric, and one topological. The lens of geometry allows us to utilise properties such as volume to understand the underlying topology. The lens of topology allows us to use a more flexible theory to dissect the underlying subcomplexes. In the former case we study geometric triangulations of hyperbolic 2--bridge link complements. Using the hyperbolic structure induced on the triangulations, we are able to establish new lower bounds on the complexity of infinitely many 2-- bridge link complements. The main tool we use for this is hyperbolic volume by way of angle structures. In the latter case, we utilise normal surface theory together with layered and 0--efficient triangulations to explore the anatomy of minimal triangulations by way of the Z2--Thurston norm. Our main result bounds the number of edges of degree three within such triangulations and hence gives us a method of analysing their subcomplexes intersecting layered solid tori. As an application, we determine new infinite families of minimal triangulations.