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

Abstract

The complex of curves $\mathcal{C}(S_g)$ of a closed orientable surface of genus $g \geq 2$ is the simplicial complex having its vertices, $\mathcal{C}^0(S_g)$, are isotopy classes of essential curves in $S_g$. Two vertices co-bound an edge of the $1$-skeleton, $\mathcal{C}^1(S_g)$, if there are disjoint representatives in $S_g$. A metric is obtained on $\mathcal{C}^0(S_g)$ by assigning unit length to each edge of $\mathcal{C}^1(S_g)$. 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 $\mathcal{C}^1 (S_g)$. Recently, Birman, Margalit and the second author introduced the concept of {\em initially efficient geodesics} in $\mathcal{C}^1(S_g)$ and used them to give a new algorithm for computing the distance between vertices. In this note we introduce the software package MICC ({\em Metric in the Curve Complex}), a partial implementation of the initially efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, we give examples of distance four vertex pairs, for $g=2$ and 3. Previously, there was only one known example, in genus $2$, due to John Hempel.