Self-intersections are empirically Gaussian

TL;DR

This paper demonstrates that the distribution of self-intersection numbers for free homotopy classes of length twenty on a punctured torus follows an approximately Gaussian distribution, based on combinatorial and computational analysis.

Contribution

It provides the first empirical evidence that self-intersection numbers on a punctured torus are distributed in a Gaussian manner for fixed word length.

Findings

Self-intersection numbers follow a Gaussian distribution.

Computed for all classes of length twenty on the punctured torus.

Histogram of self-intersections approximates a normal distribution.

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.