Trees of Nuclei and Bounds on the Number of Triangulations of the 3-Ball

Abstract

Based on the work of Durhuus-Jónsson 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 * .

The tensor track, III

“…For d ≥ 3. we do not even know whether the number of triangulations of the d-sphere with n d-simplices is exponentially bounded in n [102]. There are only limited results: bounds on locally constructible triangulations [103,104,105], or the fact that spheres are rare [106]. Progress may come from unexpected directions, such as the one of tensor models, since they do not ponder triangulations solely according to their topological class.…”

Mogami manifolds, nuclei, and 3D simplicial gravity

“…More specifically: Let us call nucleus a 3-ball with all vertices on the boundary, and in which every interior triangle contains at least two interior edges. (The notion was first introduced by Hachimori, under the name "reduced ball" [Hac00, p. 85]; the name "nucleus" appears in [CEY14]). The enumeration problem of 3-balls (or 3-spheres) is equivalent to the question of whether nuclei are exponentially many, or more [CEY14,Theorem 5.17].…”

Exponential Bounds on the Number of Causal Triangulations

“…Subsequently it was proven that not all triangulations of S 3 are locally constructible [8] and a proof of (1) is still missing. See [10] for a recent result which gives a new sufficient condition for (1) to hold. Monte Carlo simulations of the 3-dimensional gravity models indicate that a bound of the form (1) is valid, see [7,9].…”

“…Subsequently it was proven that not all triangulations of S 3 are locally constructible [8] and a proof of (1) is still missing. See [10] for a recent result which gives a new sufficient condition for…”