A new combinatorial class of 3-manifold triangulations

TL;DR

This paper introduces a new class of 3-manifold triangulations that combine weak efficiency and minimality, using twisted squares, and explores their topological implications through volume function maxima.

Contribution

It defines a novel combinatorial class of 3-manifold triangulations and analyzes their properties via twisted squares, linking triangulation features to manifold topology.

Findings

Restrictions on 3-manifold topology from volume maxima

Characterization of triangulations with weak efficiency and minimality

Application of twisted squares in triangulation analysis

Abstract

We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.