# Singular Vertices and the Triangulation Space of the D-sphere

## TL;DR

This paper presents numerical evidence that generic triangulations of D-spheres for D>3 contain singular (D-3)-simplices, with their sharing simplices increasing with volume, impacting models of quantum gravity.

## Contribution

The study reveals the presence of singular simplices in high-dimensional sphere triangulations and analyzes their behavior, providing new insights into the structure of triangulation spaces.

## Key findings

- Singular (D-3)-simplices are common in triangulations of D-spheres for D>3.
- The number of simplices sharing the singular simplex follows a power law with volume.
- Implications for the dynamical triangulation model of quantum gravity are discussed.

## Abstract

By a sequence of numerical experiments we demonstrate that generic triangulations of the $D-$sphere for $D>3$ contain one {\it singular} $(D-3)-$simplex. The mean number of elementary $D-$simplices sharing this simplex increases with the volume of the triangulation according to a simple power law. The lower dimension subsimplices associated with this $(D-3)-$simplex also show a singular behaviour. Possible consequences for the DT model of four-dimensional quantum gravity are discussed.