Saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{SL}_2(\mathbb{R})$ on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a Poincaré section for the horocycle flow.

In this talk, we will describe a Poincaré section for horocycle flow in the smallest nontrivial stratum $\mathcal{H}(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.