Self-intersections are empirically Gaussian

2010/11/28 by Moira Chas, Chas, Moira · 1 citation
Computer Science · Mathematics · #37A25 #37B10 #57M07 #Computational Geometry and Mesh Generation #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1011.6085

openalex publication_date 2010/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free homotopy class (that is, the smallest number of self-crossings of a representative of a class) can be computed in terms of the word. For each of the free homotopy classes of length twenty on the punctured torus, we compute its self-intersection number and make a histogram of how many have self-intersection 0, 1, 2..... The histogram is essentially Gaussian.

Cited by

• Experiments suggesting that the distribution of the hyperbolic length of\n closed geodesics sampling by word length is Gaussian

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