On the number of coloured triangulations of $d$-manifolds

Abstract

We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\ge 3$ is fixed. In the special case of dimension $3$, the lower and upper bounds match up to exponential factors, and we show that there are $2^{O(n)}{n}^{\frac{n}{6}}$ coloured triangulations of $3$-manifolds with $n$ tetrahedra. Our results also imply that random coloured triangulations of $3$-manifolds have a sublinear number of vertices. Our upper bounds apply in particular to coloured $d$-spheres for which they seem to be the best known bounds in any dimension $d\ge 3$, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each $3$-coloured component is planar, which is of independent interest.

15 pages. New version, proof of the lower bound corrected

References in corpus (2)

• Invitation to Random Tensors
• Mogami manifolds, nuclei, and 3D simplicial gravity

Cited by in corpus (2)

• Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model
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