The Triple Point Spectrum of closed orientable 3-manifolds

TL;DR

This paper explores the triple point spectrum as a topological invariant for closed 3-manifolds, computes it for specific manifolds, and discusses its properties as a measure of manifold complexity.

Contribution

It introduces basic properties of the triple point spectrum and computes it explicitly for  and  3-manifolds, advancing understanding of this invariant.

Findings

Computed the triple point spectrum for  and  3-manifolds.

Established basic properties of the triple point spectrum.

Presented the triple point numbers as a measure of 3-manifold complexity.

Abstract

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representaci\'on de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a measure of the complexity of a 3-manifold using the number of triple points of minimal filling Dehn surfaces. Basic properties of these invariants are presented, and the triple point spectra of $\mathbb{S}^2\times \mathbb{S}^1$ and $\mathbb{S}^3$ are computed.