Minimal triangulation size of Seifert fibered spaces with boundary

TL;DR

This paper establishes a relationship between the triangulation complexity and Seifert data of boundary-including Seifert fibered spaces, and shows how to make singular fibers simplicial through barycentric subdivision.

Contribution

It provides a formula for triangulation complexity based on Seifert data and demonstrates a method to simplify singular fibers in triangulations.

Findings

Triangulation complexity relates to Seifert data up to a constant.

All singular fibers except multiplicity two can be made simplicial.

A bound on the number of barycentric subdivisions needed.

Abstract

One measure of the complexity of a 3-manifold is its triangulation complexity: the minimal number of tetrahedra in a triangulation of it. A natural question is whether we can relate this quantity to its topology. We determine the triangulation complexity of Seifert fibered spaces with non-empty boundary in terms of their Seifert data, up to a multiplicative constant. We also show that all singular fibres of such a Seifert fibered space (aside from those of multiplicity two) can be made simplicial in the 79th barycentric subdivision of any triangulation of it.