The restriction of efficient geodesics to the nonseparating complex of curves

Seth Hovland and Greg Vinal

Vol. 330 (2024), No. 2, 233–249

DOI: 10.2140/pjm.2024.330.233

Abstract

In the complex of curves of a closed orientable surface of genus $g$, $\mathcal{𝒞}(S_g)$, a preferred finite set of geodesics between any two vertices, called efficient geodesics, was introduced by Birman, Margalit, and Menasco (2016). The main tool they used to establish the existence of efficient geodesics was a dot graph, which records the intersection pattern of a reference arc with the simple closed curves associated with a geodesic path. The idea behind the construction was that a geodesic that is not initially efficient contains shapes in its corresponding dot graph. These shapes then correspond to surgeries that reduce the intersection with the reference arc. We show that the efficient geodesic algorithm can be restricted to the nonseparating curve complex; the proof of this will involve analysis of the dot graph and its corresponding surgeries. Moreover, we demonstrate that given any geodesic in the complex of curves we may obtain an efficient geodesic whose vertices, with the possible exception of the endpoints, are all nonseparating curves.

Keywords

curve complex, nonseparating curves, efficient geodesics

Mathematical Subject Classification

Primary: 57K20

Milestones

Received: 1 October 2023

Revised: 20 June 2024

Accepted: 18 July 2024

Published: 13 September 2024