Self-intersections in combinatorial topology: statistical structure

2010/12/02 by Moira Chas, Chas, Moira, Steven P. Lalley +1 · 2 citations
Computer Science · Mathematics · #37D40 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 57M05 #Topological and Geometric Data Analysis #secondary 53C22

paper · pdf · doi:10.48550/arxiv.1012.0580

openalex publication_date 2010/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m letters, then for large m the distribution of the self-intersection number approaches the Gaussian distribution.

Cited by

• Effective equidistribution of intersection points in hyperbolic manifolds
• Singularity Cipher: A Topology-Driven Cryptographic Scheme Based on Visual Paradox and Klein Bottle Illusions

Related

• Self-intersections are empirically Gaussian
• The Eventual Gaussian Distribution for Self-Intersection Numbers on Closed Surfaces
• Self-intersection numbers of curves on the punctured torus
• Self-intersection numbers of curves in the doubly-punctured plane
• Self-Intersections of Random Geodesics on Negatively Curved Surfaces