On layered triangulations of 3-manifolds: a motivation, a generalisation, and an application

In low-dimensional topology, triangulations are used to decompose complex spaces into manageable pieces, often for the purpose of performing computations. There are many ways to triangulate a given space, and different triangulations may offer different benefits. This thesis studies the class of layered triangulations, which have appealing properties in terms of both their structure and complexity. In relation to layered triangulations, we cover: (1) their low triangulation complexity as a motivation for studying them, (2) a generalisation of their construction that leads to an algorithm to build them, and (3) their use in a related field where we can apply their structure to simplify a notoriously difficult problem.