2014/08/18 by Paul Glenn, William W. Menasco, Glenn, Paul +5 · 1 citation
Computer Science · Mathematics · #20F38 #57M60 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1408.4134
openalex publication_date 2014/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \em 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 (\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.