MICC: A tool for computing short distances in the curve complex

Abstract

The complex of curves C(Sg) of a closed orientable surface of genus g≥2 is the simplicial complex whose vertices, C0(Sg), are isotopy classes of essential simple closed curves in Sg. Two vertices co-bound an edge of the 1-skeleton, C1(Sg), if there are disjoint representatives in Sg. A metric is obtained on C0(Sg) by assigning unit length to each edge of C1(Sg). Thus, the distance between two vertices, d(v,w), corresponds to the length of a geodesic—a shortest edge-path between v and w in C1(Sg). In Birman et al. (2016), the authors introduced the concept of efficient geodesics in C1(Sg) and used them to give a new algorithm for computing the distance between vertices. In this note, we introduce the software package MICC (Metric in the Curve Complex), a partial implementation of the efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, up to an action of an element of the mapping class group, we give a calculation which produces all distance 4 vertex pairs for g=2 that intersect 12 times, the minimal number of intersections needed for this distance and genus.