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

الأصل · EN

The complex of curves C(Sg) of a closed orientable surface of genus g ≥ 2 is the simplicial complex having its vertices, C⁰(Sg), are isotopy classes of essential 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). Recently, Birman, Margalit and the second author introduced the concept of initially 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 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.