Self-intersections of curves on a surface and Bernoulli numbers

Abstract

We study an operation which measures self-intersections of curves on an oriented surface. It turns out that a certain computation on this topological operation is related to the Bernoulli numbers $B_m$, and our study yields a family of explicit formulas for $B_m$. As a special case, this family contains the celebrated formula for $B_m$ due to Kronecker.