Equivalence Relations Between Closed Curves On the Pair of Pants

TL;DR

This paper investigates a generalized equivalence relation among closed curves on a pair of pants, showing that certain k-equivalences imply simpler equivalences, and explores properties of 1-equivalence in detail.

Contribution

It proves that k-equivalence implies 1- and 2-equivalence for curves on the pair of pants, extending previous concepts and deepening understanding of 1-equivalence properties.

Findings

k-equivalence implies 1- and 2-equivalence

Properties of 1-equivalence are characterized

Generalization of Leininger's concept is established

Abstract

We examine an equivalence relation between free homotopy classes of closed curves on the pair of pants known as k-equivalence, a generalization of a concept previously defined by Leininger. We prove that two classes of closed curves on the pair of pants that are k-equivalent must also be 1-equivalent and 2-equivalent. We also examine properties of 1-equivalence on the pair of pants in greater depth.