Annular-Efficient Triangulations of 3-manifolds

TL;DR

This paper introduces annular-efficient triangulations for 3-manifolds, characterizing their properties and showing how to modify any triangulation to achieve this efficiency, with implications for boundary slopes of certain surfaces.

Contribution

It defines annular-efficient triangulations, proves their existence for certain 3-manifolds, and explores their implications for boundary slopes of incompressible surfaces.

Findings

Annular-efficient triangulations imply irreducibility and boundary-irreducibility.

Any triangulation of such manifolds can be modified to be annular-efficient.

Finite boundary slopes exist for surfaces with bounded Euler characteristic.

Abstract

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, irreducible, boundary-irreducible, and an-annular 3-manifold, any triangulation can be modified to an annular-efficient triangulation. It follows that for a manifold satisfying this hypothesis, there are only a finite number of boundary slopes for incompressible and boundary-incompressible surfaces of a bounded Euler characteristic.