Self-intersections of arcs on a pair of pants

TL;DR

This paper studies arcs on a pair of pants, providing an algorithm to compute their self-intersection numbers, establishing bounds related to word length, and showing the spectrum of these numbers covers all natural numbers.

Contribution

It introduces an algorithm for calculating self-intersection numbers of arcs and explores their spectrum, offering new bounds and comprehensive coverage of natural numbers.

Findings

Algorithm for self-intersection computation

Bounds for self-intersection in terms of word length

Spectrum covers all natural numbers

Abstract

We investigate arcs on a pair of pants and present an algorithm to compute the self-intersection number of an arc. Additionally, we establish bounds for the self-intersection number in terms of the word length. We also prove that the spectrum of self-intersection numbers of 2-low-lying arcs covers all natural numbers.