The Saddle Connection Graph of the Golden L

Abstract:

The saddle connection graph of a translation surface is a graph with saddle connections as vertices and sets of two disjoint saddle connections as edges. It characterizes translation surfaces up to affine equivalence. The graph of slopes summarizes saddle connections with the same slope in equivalence classes. With the help of an algorithm that generates all directions of saddle connections and some findings that reduce the number of pairwise comparisons to find the edges of the graph, we construct a finite approximation of the graph of slopes of a finite translation surface, which is called the golden L.