Trees of nuclei and bounds on the number of triangulations of the 3-ball

TL;DR

This paper introduces the concept of nuclei in 3-ball triangulations and demonstrates that all triangulations can be constructed from trees of these nuclei, providing new bounds on their enumeration.

Contribution

It defines nuclei as a fundamental building block for 3-ball triangulations and links their enumeration to bounds on the total number of triangulations.

Findings

Every triangulation can be constructed from trees of nuclei.

If nuclei with t tetrahedra are bounded by C^t, then total triangulations are bounded by C_*^t.

Provides a new reformulation of Gromov's question on triangulation bounds.

Abstract

Based on the work of Durhuus-J{\'o}nsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can be built from trees of nuclei. This leads to a new reformulation of Gromov's question: We show that if the number of rooted nuclei with $t$ tetrahedra has a bound of the form $C^t$, then the number of rooted triangulations with $t$ tetrahedra is bounded by $C_*^t$.