Self-intersections are empirically Gaussian

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.