Intersections of multicurves from Dynnikov coordinates

TL;DR

This paper introduces an algorithm that computes the geometric intersection number of multicurves on an n-punctured disk using Dynnikov coordinates, with a specific complexity bound, leveraging a multicurve relaxation algorithm.

Contribution

It provides a new algorithm for calculating intersection numbers from Dynnikov coordinates with proven complexity, improving computational methods in geometric topology.

Findings

Algorithm computes intersection numbers efficiently

Complexity is $O(m^2n^4)$ based on coordinate sum

Uses Cumplido's multicurve relaxation algorithm

Abstract

We present an algorithm for calculating the geometric intersection number of two multicurves on the $n$-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity $O(m^2n^4)$, where $m$ is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredient is an algorithm due to Cumplido for relaxing a multicurve.