Intersection numbers in the curve complex via subsurface projections

TL;DR

This paper establishes a new way to measure intersection numbers of curves in the curve complex using subsurface projections, leading to an algorithm for computing distances between curves.

Contribution

It introduces a converse inequality relating intersection numbers to subsurface projection distances and develops an algorithm for curve distance determination.

Findings

Intersection number can be measured by subsurface projection distances.

A coarse decreasing property of intersection numbers in tight multigeodesics.

An algorithm for determining curve distances in the curve complex.

Abstract

A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of two curves can be measured by the sum of all subsurface projection distances between them. As an application of this result, we obtain a coarse decreasing property of the intersection numbers of the multicurves contained in tight multigeodesics. Furthermore, by using this property, we give an algorithm for determining the distance between two curves in the curve complex. Indeed, such algorithms have been also found by Birman--Margalit--Menasco, Leasure, Shackleton, and Webb: we will briefly compare our algorithm with some of their algorithms, for detailed quantitative comparison of all known algorithms including our algorithm, we refer the reader to the paper of Birman--Margalit--Menasco \cite{BMM}.