Distance and intersection number in the curve graph of a surface

Abstract

In this work, we study the cellular decomposition of S induced by a filling pair of curves v and w, \(Dec_{v,w}(S) = S {\setminus } (v \cup w)\), and its connection to the distance function d(v, w) in the curve graph of a closed orientable surface S of genus g. Building on the work of Leasure, efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in 2016, giving an algorithm that begins with a pair of non-separating filling curves that determine vertices (v, w) in the curve graph of a closed orientable surface S and computing from them a finite set of efficient geodesics. We extend the tools of efficient geodesics to study the relationship between distance d(v, w), intersection number i(v, w), and \(Dec_{v,w}(S)\). The main result is the development and analysis of particular configurations of rectangles in \(Dec_{v,w}(S)\) called spirals. We are able to show that, with appropriate restrictions, the efficient geodesic algorithm can be used to build an algorithm that reduces i(v, w) while preserving d(v, w). At the end of the paper, we note a connection between our work and the notion of extending geodesics.