Library of Triangulations

Random discrete Morse theory and a new library of triangulations. The discrete Morse theory, as defined by R. Forman, is used in a random algorithm. This means that the function is chosen randomly, and then the number of critical points is calculated leading to the random discrete Morse vector (c 0 ,⋯,c d ). This is considered as a kind of measuring the complexity of the triangulation. The discrete Morse vector can be much larger than the Betti vector but it turns out that in many cases the random Morse function produces the optimal Morse vector, i.e., the function is perfect. This is illustrated by numerous computer experiments with 10.000 choices of Morse functions on concrete triangulations of certain manifolds with a number of vertices ranging between 8 and several thousands. Typically most of the functions were perfect, in some cases more than 90%. This percentage can be used as an indicator for the complexity of a triangulation. A particular consequence is that all 250.359 triangulated 3-manifolds with up to 10 vertices admit a perfect discrete Morse function