Efficient geodesics and an effective algorithm for distance in the complex of curves

TL;DR

This paper introduces a new, simple, and more effective algorithm for computing distances in the complex of curves, along with a novel set of efficient geodesics that differ from existing tight geodesics.

Contribution

The paper presents a new algorithm and a new class of efficient geodesics for the complex of curves, improving computational effectiveness over prior methods.

Findings

Algorithm efficiently computes distances for all accessible cases

Introduces a finite set of efficient geodesics

Demonstrates improved effectiveness over previous algorithms

Abstract

We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.