Berlin 2015 – scientific programme

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 16: Gravitation

MP 16.2: Talk

Thursday, March 19, 2015, 11:40–12:00, HFT-FT 101

Counting of Manifold Triangulations — •Benedikt Krüger and Klaus Mecke — Institut für Theoretische Physik, Staudtstr. 7, 91058 Erlangen

Each topological manifold in 2d and 3d permits a finite number of non-equivalent discretisations into combinatorial manifolds or triangulations with given number of vertices or maximal simplices. This number of distinct triangulations is important for questions arising in topology, geometry and physics. E.g., the scaling behaviour of this number determines whether the quantum gravity model of causal dynamical triangulations [1] is well-defined.

Until now the best method for counting of combinatorial manifolds was the isomorphism free enumeration of all possible triangulations for vertex numbers below 15 [2]. Here, we use Monte-Carlo algorithms for estimating the number of triangulations of two- and three-dimensional manifolds and show that the accessible regime of triangulation counts can be increased by several magnitudes. We give numerical evidence that the number of surface triangulations scales exponentially with the vertex number and that the rate of growth depends linearly on the genus of the surface. Additionally we address the question whether the number of triangulations of the 3-sphere scales exponentially with the number of tetrahedra, and whether these triangulations are computationally ergodic.

[1] J. Ambjörn, J. Jurkiewicz, and R. Loll, Phys. Rev. D 72, 064014 (2005); [2] T. Sulanke and F. H. Lutz, Eur. J. Comb. 30, 1965 (2009)