Counting intersections of normal curves

TL;DR

This paper introduces a fast algorithm for counting intersections of normal curves on triangulated surfaces, facilitating the analysis of mapping class groups through matrix representations and an exotic multiplication, and offers an efficient solution to the word problem.

Contribution

It presents a novel fast algorithm for intersection counting and a new matrix-based approach for mapping class groups of punctured surfaces.

Findings

Efficient intersection counting algorithm for normal curves.

Matrix representation of mapping class groups.

Solution to the word problem with improved efficiency.

Abstract

A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class groups of punctured surfaces is proposed, with efficiency understood in a more restrictive way than the most common one.